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Entropy of Bernoulli Convolutions and Uniform Exponential Growth for Linear Groups
Abstract
The exponential growth rate of non polynomially growing subgroups of
is conjectured to admit a uniform lower bound. This is known for non-amenable
subgroups, while for amenable subgroups it is known to imply the Lehmer
conjecture from number theory. In this note, we show that it is equivalent to
the Lehmer conjecture. This is done by establishing a lower bound for the
entropy of the random walk on the semigroup generated by the maps , where is an algebraic number. We give a bound
in terms of the Mahler measure of . We also derive a bound on the
dimension of Bernoulli convolutions.
... Breuillard and the author [15] gave an estimate for the entropy of the random walk on the semigroup generated by S λ in terms of M λ . The proof utilizes measures related to Bernoulli convolutions. ...
... The quantity h λ has been studied by Breuillard and the author [15]. They gave the following bounds in terms of the Mahler measure M λ . ...
... In order that we can conclude that ν λ is absolutely continuous, we need to find a rate for the speed of convergence in (15). Indeed, Garsia [24] observed that ν λ is absolutely continuous if the sequence d − H(ν λ ; 2 −d ) is bounded. ...
The Bernoulli convolution with parameter is the measure on that is the distribution of the random power series , where are independent fair coin-tosses. This paper surveys recent progress on our understanding of the regularity properties of these measures.
... Further recent progress was made by Breuillard and Varju [3], where it was proved that H(β) ≥ 0.44 min{log 2, log M β }, for any algebraic integer β ∈ (1, 2), where H(β) is the Garsia entropy of ν β (see Section 1.1 for the definition) and M β is the Mahler measure of β defined by M β = |β i |>1 |β i |, where β i are the algebraic conjugates (including β itself) of β. This implies that for an algebraic integer β ∈ (1, 2), dim H (ν β ) = 1 if 0.44 min{log 2, log M β } ≥ log β (see (1)). ...
... see also [3] for a more detailed explanation. ...
... We conclude that dim H (ν β ) = 1. We remark that this result does not follow from the aforementioned work of Breuillard and Varju [3], since in this example 0.44 min{log 2, log M β } log β ≈ 0.6146 < 1. ...
We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution to arbitrary given accuracy whenever is algebraic. In particular, if the Garsia entropy is not equal to then we have a finite time algorithm to determine whether or not .
... The dimension of Bernoulli convolutions for algebraic parameters has been studied in the paper [3]. Recall that Lehmer's conjecture states that there is some numerical constant ε 0 ą 0 such that the Mahler measure M λ (the definition is recalled below in p1.6q) of every algebraic number λ is either 1 or at least 1`ε 0 . ...
... Recall that Lehmer's conjecture states that there is some numerical constant ε 0 ą 0 such that the Mahler measure M λ (the definition is recalled below in p1.6q) of every algebraic number λ is either 1 or at least 1`ε 0 . It was proved in [3] that Lehmer's conjecture implies that there exists a number a ă 1 such that dim µ λ " 1 for all algebraic numbers λ P pa, 1q. We can now drop the condition of algebraicity in that result thanks to Corollary 3 and we obtain the following. ...
... where Hp¨q denotes the Shannon entropy of a discrete random variable. With this notation Hochman's formula is The quantity h λ has been studied in the paper [3]. It was proved there [3, Theorem 5] that there is an absolute constant c 0 ą 0 such that for any algebraic number, we have ...
The Bernoulli convolution with parameter is the probability measure that is the law of the random variable , where the signs are independent unbiased coin tosses. We prove that each parameter with can be approximated by algebraic parameters within an error of order for any number A, such that . As a corollary, we conclude that for each of . These are the first explicit examples of such transcendental parameters. Moreover, we show that Lehmer's conjecture implies the existence of a constant such that for all .
... (See [4,Section 3.4], where the formula is derived in this form from Hochman's main result.) The quantity h λ has been studied in the paper [4]. ...
... (See [4,Section 3.4], where the formula is derived in this form from Hochman's main result.) The quantity h λ has been studied in the paper [4]. It was proved there [4,Theorem 5] that there is an absolute constant c 0 > 0 such that for any algebraic number, we have c 0 · min(log M λ , 1) ≤ h λ ≤ min(log M λ , 1). ...
... The quantity h λ has been studied in the paper [4]. It was proved there [4,Theorem 5] that there is an absolute constant c 0 > 0 such that for any algebraic number, we have c 0 · min(log M λ , 1) ≤ h λ ≤ min(log M λ , 1). ...
We prove that Bernoulli convolutions are absolutely continuous provided the parameter lambda is an algebraic number sufficiently close to 1 depending on the Mahler measure of lambda.
... The second component is the connection, obtained by Breuillard and Varjú [9], between random walk entropy and Mahler measure. For an affine IFS Ψ = {ψ i } i∈Λ , we write h RW (Ψ, p) for the entropy of the random walk generated by Ψ and p (see Section 2.4.4). ...
... The definition of the Mahler measure of an algebraic number is provided in Section 5.1. For (η 1 , ..., η d ) = η ∈ Ω such that η j0 is algebraic, it follows from the results of [9] that h RW (Φ η j0 , p) is close to its maximal possible value H(p) whenever the Mahler measure of η j0 is sufficiently large (see Theorem 5.1). Here, Φ η j0 := {t → η j0 t + a i,j0 } i∈Λ . ...
... In [8], the inequality h RW (Φ η , p) < κ + ϵ is replaced with dim µ η < dim µ λ + ϵ, where µ η is the Bernoulli convolution associated to η. When d = 1 and dim µ λ + ϵ/χ 1 < 1, it follows by the work of Hochman that these inequalities are equivalent (see [9,Section 3.4]). The intuition behind the value κ is best explained by [30,Lemma 4.1], in which it is shown that κ equals the limit of the normalized entropies 1 n H (µ λ , E n ), where E n is the level-n non-conformal partition of R d determined by λ (see Section 2.6). ...
For with , denote by the Bernoulli convolution associated to . That is, is the distribution of the random vector , where the signs are chosen independently and with equal weight. Assuming for each that is not a root of a polynomial with coefficients , we prove that the dimension of equals , where is the Lyapunov dimension. More generally, we obtain this result in the context of homogeneous diagonal self-affine systems on with rational translations. The proof extends to higher dimensions the works of Breuillard and Varj\'u and Varj\'u regarding Bernoulli convolutions on the real line. The main novelty and contribution of the present work lies in an extension of an entropy increase result, due to Varj\'u, in which the amount of increase in entropy is given explicitly. The extension of this result to the higher-dimensional non-conformal case requires significant new ideas.
... See [9,Section 3.4] for the details of how the following follows from the main result of Hochman [23]. ...
... This quantity is widely used in number theory as a measure of the "complexity" of Á. Notice that if Á 2 Q, then M.Á/ is the maximum of the absolute values of the numerator and the denominator of Á. [9] found a connection between the entropy rate and the Mahler measure. A form of this most suited for the proof of Theorem 7 is the following. ...
... See [54,Theorem 9] for the details of how this follows from the technical results of [9]. Using this theorem, we conclude that M.Á n / < C for a constant C that only depends on , but not on n. ...
... Varjú's proof operates by using a result of Breuillard and Varjú that transcendental λ with dim(ν λ ) < 1 can be approximated arbitrarily well by algebraic λ n with h λn (ν) < log λ −1 −ε, with ε depending on λ [5]. A crucial part of the proof is to bound the Mahler measure of those λ n , which is possible due to the result of Breulliard and Varjú [4] that for every ε there is C = C(ε) such that ...
... Even though these are somewhat special examples and we know that absolute continuity and power decay hold generically, there are few other concrete examples. In the case of absolute continuity, Varjú [4] showed that there is a very small c such that for all algebraic λ with λ > 1 − c min {log M λ , (log M λ ) −1.1 }, the measure ν Ber λ is absolutely continuous. Kittle [15] generalised the ideas to show that for algebraic λ with Mahler measure close to two, the measure is also absolutely continuous. ...
... This space is naturally associated to λ and is a product of R s with some non-Archimedean fields. The measure µ λ for λ an algebraic unit is the one considered in the papers [4], [2] to calculate the Garsia entropy (see "Computation of the Garsia entropy"); their method works exactly when A λ ∼ = R s and no non-Archimedean fields are present. We introduce both µ λ and A λ formally in Subsection 1.3. ...
In this paper, we consider the self-similar measure on , where and the are independent, identically distributed with respect to a measure finitely supported on . One example of this is the classical Bernoulli convolution. It is known that for certain combinations of algebraic and uniform on an interval, is absolutely continuous and its Fourier transform has power decay (\cite{garsia1}, \cite{feng}); in the proof, it is exploited that for these combinations, a quantity called the Garsia entropy is maximal. We show that absolute continuity and power Fourier decay occur when and are such that is maximal and classify all combinations for which this is the case. We find that if an algebraic without a Galois conjugate of modulus exactly one has a such that is maximal, then all Galois conjugates of must be smaller in modulus than one and must satisfy a certain finite set of linear equations in terms of .
... Let ε ∈ (0, 1). Then for all but an ε-fraction of all a ∈ F × p , the mixing times of the Markov chain (1.1) satisfy for any δ ∈ (0, 1 2 ): ...
... Our motivation for studying the Markov chain (1.1) is manifold. First of all this chain is the natural "mod p analogue" of the classical Bernoulli convolutions we studied in [1], where a close relationship between the entropy of the random walk and the Mahler measure of the multiplier was established. This enabled us to give a "mod p" reformulation of the Lehmer conjecture in [3]. ...
... Supp(μ (n) a ) = F p for all n > 2T 2 (1). In other words D a ( p) 2T 2 (1) + 1. ...
We study the Markov chain on a finite field , where is fixed and are independent and identically distributed random variables in . Conditionally on the Riemann hypothesis for all Dedekind zeta functions, we show that the chain exhibits a cut-off phenomenon for most primes p and most values of . We also obtain weaker, but unconditional, upper bounds for the mixing time.
... See [9,Section 3.4] for the details of how the following follows from the main result of Hochman [23]. ...
... Breuillard and Varjú [9] found a connection between the entropy rate and the Mahler measure. A form of this most suited for the proof of Theorem 7 is the following. ...
... See [54,Theorem 9] for the details of how this follows from the technical results of [9]. ...
We discuss the problem of determining the dimension of self-similar sets and measures on . We focus on the developments of the last four years. At the end of the paper, we survey recent results about other aspects of self-similar measures including their Fourier decay and absolute continuity.
... a n give rise to the same sum defined after normalizing by log(β). It follows from the definition that if β is not a height 1 algebraic integer then H(β) = log (2), and hence dim H (ν β ) = 1. Very recently Varju has shown that if β is non-algebraic then dim H (ν β ) = 1 [13]. ...
... Since this line β 1 = β 2 causes a particular problem for our techniques, it would be useful to know whether the constant c in Theorem 5 of [2], for which an approximate value of 0.44 was given, could be improved to give a constant larger than 0.5. In that case one would automatically have that algebraic β with Galois conjugate β 2 of similar modulus to β have dimension one, and so values close to the problematic line β 1 = β 2 would be solved by other methods. ...
... In this section we will discuss the case where β 1 > 1 has a real conjugate β 2 with |β 2 | > 1. Corollaries 2.4 and 3.3 give two curves in the square [1,2] × [1,2] such that for any pair of Galois conjugates (β 1 , β 2 ) above these curves we have dim H (ν β 1 ) = 1 (provided β 1 = β 2 ). We have indicated the curves coming from For points below these curves, we do a computational check using Proposition 2.2. ...
We introduce a parameter space containing all algebraic integers β∈(1,2] that are not Pisot or Salem numbers, and a sequence of increasing piecewise continuous function on this parameter space which gives a lower bound for the Garsia entropy of the Bernoulli convolution νβ. This allows us to show that dimH(νβ)=1 for all β with representations in certain open regions of the parameter space.
... On the other hand, Breuillard and Varjú [3] gave an estimate on h(λ) in terms of the Mahler measure M λ : ...
... The upper bound in (19) is often strict. In particular, it is known that h(λ) < M λ , provided λ has no Galois conjugates on the unit circle [3]. ...
... Then ψ α,T S µ = D (2) α,S ψ α,µ . 3 One possible approach is via the Gamma function. We rewrite |s| −α = 2π α/2 Γ(α/2) ∞ 0 t α−1 e −πt 2 s 2 dt and compute the Fourier transform of the latter by swapping the order of integrals. ...
In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The Bernoulli convolution measure is the probability measure corresponding to the law of the random variable , where are i.i.d. random variables assuming values and 1 with equal probability and . In particular, for Bernoulli convolutions we give a uniform lower bound for all .
... Another key ingredient in [22] is the main result of [5], which provides an estimate for the Mahler measure of a parameter λ if the corresponding Bernoulli convolution has lots of exact overlaps. Mahler measure is a widely used quantity in number theory to measure the complexity of an algebraic number. ...
... An analogue of the main result of [5] would give an affirmative answer to the following question. Question 1.5. ...
... We do not know the answer to Question 1.5, but we can easily deduce the following weaker statement from the results of [5]. Theorem 1.7. ...
We study the dimension of self-similar measures associated to a homogeneous iterated function system of three contracting similarities on and other more general IFS's. We extend some of the theory recently developed for Bernoulli convolutions to this setting. In the setting of three maps a new phenomenon occurs, which has been highlighted by recent examples of Baker, and B\'ar\'any, K\"aenm\"aki. To overcome the difficulties stemming form these, we develop novel techniques, including an extension of Hochman's entropy increase method to a function field setup.
... a n give rise to the same sum defined after normalizing by log(β). It follows from the definition that if β is not a height 1 algebraic integer then H(β) = log (2), and hence dim H (ν β ) = 1. Very recently Varju has shown that if β is non-algebraic then dim H (ν β ) = 1 [10]. ...
... Since this line β 1 = β 2 causes a particular problem for our techniques, it would be useful to know whether the constant c in Theorem 5 of [2], for which an approximate value of 0.44 was given, could be improved to give a constant larger than 0.5. In that case one would automatically have that algebraic β with Galois conjugate β 2 of similar modulus to β have dimension one, and so values close to the problematic line β 1 = β 2 would be solved by other methods. ...
... With this in mind, Figure 4.1 is a graph of points (β 1 , β 2 ) such that if a polynomial p(x) has roots "close to" β 1 and β 2 then dim H (ν β ) = 1. We have a large region [1,2] × [1,2] for which we know the dim H (ν β ) = 1 by Corollary 3.3. An additonal subregion of [1,2] × [1,2] is given by [8,Theorem 6.6] which we quote below. ...
We introduce a parameter space containing all algebraic integers that are not Pisot or Salem numbers, and a sequence of increasing piecewise continuous function on this parameter space which gives a lower bound for the Garsia entropy of the Bernoulli convolution . This allows us to show that for all with representations in certain open regions of the parameter space.
... Let ε P p0, 1q. Then for all but an ε-fraction of all a P F p , the mixing times of the Markov chain p1.1q satisfy for any δ P p0, 1 2 q: ...
... Our motivation for studying the Markov chain p1.1q is manifold. First of all this chain is the natural "mod p analogue" of the classical Bernoulli convolutions we studied in [1], where a close relationship between the entropy of the random walk and the Mahler measure of the multiplier was established. This enabled us to give a "mod p" reformulation of the Lehmer conjecture in [3]. ...
... Consider the Markov chain p1.1q for a residue a P Fp with multiplicative order at least C log pplog log pq. Then for every δ P p0, 1 2 q, we have ...
We study the Markov chain on a finite field , where is fixed and are independent and identically distributed random variables in . Conditionally on the Riemann hypothesis for all Dedekind zeta functions, we show that the chain exhibits a cut-off phenomenon for most primes p and most values of . We also obtain weaker, but unconditional, upper bounds for the mixing time.
... The dimension of Bernoulli convolutions for algebraic parameters is not fully understood. Hochman [9] expressed the dimension in terms of the so-called Garsia entropy of λ, a quantity that have been studied recently in [1,4]. We will briefly recall these results in Section 1.4. ...
... The algebraic number ξ in the statement is found using a characterization of parameters with dim ν λ < 1 by Breuillard and Varjú [3]. The Mahler measure of ξ is estimated using another paper [4] of the same authors. Then the conclusion of the above statement is plugged into a result of Hochman [9] to prove dim ν λ = 1. ...
... See [4,Section 3.4], where this is formally deduced from the main result of Hochman [9]. Theorem 8 reduces Question 2 for algebraic parameters to determining when h λ ≥ log λ −1 holds. ...
The Bernoulli convolution with parameter is the probability measure supported on that is the law of the random variable , where the are independent fair coin-tosses. We prove that for all transcendental .
... Further recent progress was made by Breuillard and Varju [3], where it was proved that H(β) ≥ 0.44 min{log 2, log M β }, for any algebraic integer β ∈ (1, 2), where H(β) is the Garsia entropy of ν β (see Section 1.1 for the definition) and M β is the Mahler measure of β defined by M β = |β i |>1 |β i |, where β i are the algebraic conjugates (including β itself) of β. This implies that for an algebraic integer β ∈ (1, 2), dim H (ν β ) = 1 if 0.44 min{log 2, log M β } ≥ log β (see (1)). ...
... see also [3] for a more detailed explanation. ...
... We conclude that dim H (ν β ) = 1. We remark that this result does not follow from the aforementioned work of Breuillard and Varju [3], since in this example 0.44 min{log 2, log M β } log β ≈ 0.6146 < 1. ...
We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution to arbitrary given accuracy whenever is algebraic. In particular, if the Garsia entropy is not equal to then we have a finite time algorithm to determine whether or not .
... Breuillard and the author [15] gave an estimate for the entropy of the random walk on the semigroup generated by S λ in terms of M λ . The proof utilizes measures related to Bernoulli convolutions. ...
... The quantity h λ has been studied by Breuillard and the author [15]. They gave the following bounds in terms of the Mahler measure M λ . ...
... In order that we can conclude that ν λ is absolutely continuous, we need to find a rate for the speed of convergence in (15). Indeed, Garsia [24] observed that ν λ is absolutely continuous if the sequence d − H(ν λ ; 2 −d ) is bounded. ...
The Bernoulli convolution with parameter is the measure on that is the distribution of the random power series , where are independent fair coin-tosses. This paper surveys recent progress on our understanding of the regularity properties of these measures.
... (See [2,Section 3.4], where the formula is derived in this form from Hochman's main result.) ...
... The quantity h λ has been studied in the paper [2]. It was proved there [2,Theorem 4] that there is an absolute constant c 0 > 0 such that for any algebraic number, we have ...
... The quantity h λ has been studied in the paper [2]. It was proved there [2,Theorem 4] that there is an absolute constant c 0 > 0 such that for any algebraic number, we have ...
We prove that Bernoulli convolutions are absolutely continuous provided the parameter lambda is an algebraic number sufficiently close to 1 depending on the Mahler measure of lambda.
... Many previous works have noted the advantageous properties of entropy in somewhat related settings. To give a few examples, in rough chronological order there is the work of Avez [1] and of Kaȋmanovich and Vershik [13] on random walks on discrete groups, the work of Hochman [12] on fractals, and the work of Breuillard-Varjú [3] on Bernouilli convolutions. ...
... Also, by (A. 3), ...
The entropic doubling of a random variable X taking values in an abelian group G is a variant of the notion of the doubling constant of a finite subset A of G, but it enjoys somewhat better properties; for instance, it contracts upon applying a homomorphism. In this paper we develop further the theory of entropic doubling and give various applications, including: (1) A new proof of a result of P\'alv\"olgyi and Zhelezov on the ``skew dimension'' of subsets of with small doubling; (2) A new proof, and an improvement, of a result of the second author on the dimension of subsets of with small doubling; (3) A proof that the Polynomial Freiman--Ruzsa conjecture over implies the (weak) Polynomial Freiman--Ruzsa conjecture over .
... Self-similar sets and measures are natural and important objects in fractal geometry, at the interface of geometric measure theory, ergodic theory, number theory and harmonic analysis (see the recent surveys [14,17,33]). Spectacular and influential advances in the dimension theory of these objects have been achieved in the recent period [3,12,15,18,19,34,38,39], in particular in connection with the resolutions of Furstenberg's conjectures on the Hausdorff dimension of the sums and intersections of ×2and ×3-invariant sets on the 1-dimensional torus. ...
... which is called the Garsia entropy of μ. Since Φ L is an algebraic IFS, it was implicitly proved in [15] that [3] for more explanations.) Now applying the second inequality in (5.5) to the definition of h G (μ) yields immediately that h G (μ) ≥ log N L − log κ L and so (5.14) follows from (5.15). ...
Let μ be a self-similar measure generated by an IFS Φ={ϕi}i=1ℓ of similarities on Rd (d≥1). When Φ is dimensional regular (see Definition 1.1), we give an explicit formula for the Lq-spectrum τμ(q) of μ over [0, 1], and show that τμ is differentiable over (0, 1] and the multifractal formalism holds for μ at any α∈[τμ′(1),τμ′(0+)]. We also verify the validity of the multifractal formalism of μ over [τμ′(∞),τμ′(0+)] for two new classes of overlapping algebraic IFSs by showing that the asymptotically weak separation condition holds. For one of them, the proof appeals to the recent result of Shmerkin (Ann. Math. (2) 189(2):319–391, 2019) on the Lq-spectrum of self-similar measures.
... a) In the original version of the theorem in [19], both the convolution and the translations take place on the circle [0, 1) with addition modulo 1. See [17, Theorem 2.2 and Remark 2.3] for this formulation. b) The main claim in the theorem is part (5). Obtaining sets A, B satisfying (A1)-(A4) and (B1)-(B4) is not hard, and (6) is a straightforward calculation using (5). ...
... b) The main claim in the theorem is part (5). Obtaining sets A, B satisfying (A1)-(A4) and (B1)-(B4) is not hard, and (6) is a straightforward calculation using (5). c) The theorem fails for q = 1 and q = ∞. ...
We present a self-contained proof of a formula for the dimensions of self-similar measures on the real line under exponential separation (up to the proof of an inverse theorem for the norm of convolutions). This is a special case of a more general result of the author from [Shmerkin, Pablo. On Furstenberg's intersection conjecture, self-similar measures, and the norms of convolutions. Ann. of Math., 2019], and one of the goals of this survey is to present the ideas in a simpler, but important, setting. We also review some applications of the main result to the study of Bernoulli convolutions and intersections of self-similar Cantor sets.
... In [BV16] we have shown that the Mahler measure is related to the growth rate of the cardinality of the set One can take c = 0.44. The limit above always exists by sub-multiplicativity of d → |S d (α)|. ...
... We recall the following fairly well-known lemma (see Lemma 16 in [BV16] and §8 in [B07]). ...
We give a reformulation of the Lehmer conjecture about algebraic integers in terms of a simple counting problem modulo p.
... The dimension of Bernoulli convolutions for algebraic parameters has been studied in the paper [2]. In particular, it was proved that Lehmer's conjecture implies that there exists a number a ă 1 such that dim µ λ " 1 for all algebraic numbers λ P pa, 1q. ...
... where Hp¨q denotes the Shannon entropy of a discrete random variable. With this notation Hochman's formula is The quantity h λ has been studied in the paper [2]. It was proved there [2, Theorem 5] that there is an absolute constant c 0 ą 0 such that for any algebraic number, we have ...
The Bernoulli convolution with parameter is the probability measure that is the law of the random variable , where the signs are independent unbiased coin tosses. We prove that each parameter with can be approximated by algebraic parameters within an error of order for any number A, such that . As a corollary, we conclude that for each of . These are the first explicit examples of such transcendental parameters. Moreover, we show that Lehmer's conjecture implies the existence of a constant such that for all .
... Very recently, some striking progress on the dimensions and absolute continuity of Bernoulli convolutions for algebraic parameters was achieved by P. Varjú [50] and E. Breuillard and P. Varjú [7]. The latter article also uncovers some deep connections between Bernoulli convolutions, the famous Lehmer's conjecture from number theory, and the growth of subgroups of linear groups. ...
We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the -dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of and -invariant sets. Among several other applications, we also show that Bernoulli convolutions have an density for all finite q, outside of a zero-dimensional set of exceptions. The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to norms, and likewise relies on an inverse theorem for the decay of norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemer\'{e}di-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.
... The proof of Theorem 1.1 is an outgrowth of methods from [16,17], see also [12], which dealt with the dimension of self-similar measures in Euclidean space. The stationary measure ν is in many respects like a self-similar measure: when µ is finitely supported, stationarity implies that ν decomposes into "copies" of itself via ν = A∈supp µ µ(A) · Aν. ...
Let be a measure on generating a non-compact and totally irreducible subgroup, let denote its Lyapunov exponent, and let be the associated stationary (Furstenberg) measure for the action on the projective line. We prove that if is supported on finitely many matrices with algebraic entries, then where is the random walk entropy of , and denotes pointwise dimension. In particular, for every , there is a neighborhood U of the identity in such that if a measure is supported on algebraic matrices with all atoms of size at least , and generates a group which is non-compact and totally irreducible, then its stationary measure satisfies .
... If instead of asking for absolute continuity of ν β we ask whether dim H (ν β ) = 1 then a lot more is known, mainly stemming from work of Hochman [10]. Several recent articles give conditions under which the Bernoulli convolution associated to an algebraic β has dimension one [5,4,9] or show that the Hausdorff dimension can be computed [1]. Most significantly, Varjú has shown that dim H (ν β ) = 1 whenever β is transcendental [19]. ...
We show how to turn the question of the absolute continuity of Bernoulli convolutions into one of counting the growth of the number of overlaps in the system. When the contraction parameter is a hyperbolic algebraic integer, we turn this question of absolute continuity into a question involving the ergodic theory of cocycles over domain exchange transformations.
... • [9], [2], [3], [20]: If dim H µ λ < 1 then λ is a root of infinitely many polynomials with coefficients ±1, 0. • [1]: There is an effective algorithm to approximate dim H µ λ for all algebraic λ. 1 Continuity. Let λ be an algebraic number over Q. ...
Let be an algebraic integer with Mahler measure 2. A classical result of Garsia shows that the Bernoulli convolution is absolutely continuous with respect to the Lebesgue measure with a density function in . In this paper, we show that the density function is continuous.
... So far Pisot numbers in (1,2) are the only known examples of parameters β with dim(µ β ) < 1. As for other algebraic numbers, it is known that dim(µ β ) = 1 if β is an algebraic number which is not a root of a polynomial of coefficients 0 and ±1 [17], or β is an algebraic number with relatively large Mahler measure [4], or β is among some concrete examples of algebraic numbers with small degree [1,15]. ...
In this paper, we provide an algorithm to estimate from below the dimension of self-similar measures with overlaps. As an application, we show that for any , the dimension of the Bernoulli convolution satisfies which improves a previous uniform lower bound 0.82 obtained by Hare and Sidorov \cite{HareSidorov2018}. This new uniform lower bound is very close to the known numerical approximation for , where is the largest root of the polynomial . Moreover, the infimum is attained at a parameter in a small interval When is a Pisot number, we express in terms of the measure-theoretic entropy of the equilibrium measure for certain matrix pressure function, and present an algorithm to estimate from above as well.
... Erdős showed that ν β is singular when β is a Pisot number [8], and indeed Garsia showed that such Bernoulli convolutions have dimension less than one [12]. There has been very substantial progress on the dimension theory of Bernoulli convolutions in the last decade, stemming from the work of Hochman [17], and in particular it is now known that non-algebraic β give rise to Bernoulli convolutions of dimension one [25], whereas for algebraic β there are algorithms to determine whether or not ν β has dimension one [5,2]. For a summary of recent research into the dimension theory of Bernoulli Convolutions see [24]. ...
Given a real number beta > 1, the spectrum of beta is a well studied dynamical object. In this article we show the existence of a certain measure on the spectrum of beta related to the distribution of random polynomials in beta, and discuss the local structure of this measure. We also make links with the question of the Hausdorff dimension of the corresponding Bernoulli Convolution
... See also [Fur] by Furstenberg, [Hoc4] by Hochman, [PS1] by by Shmerkin for more interesting questions on families of overlapping IFS. These problems are already very difficult in the simple case of Bernoulli convolutions, see for example [BV1,BV2,Hoc1,LPS,Shm1,Sol3,SS2,Var1,Var2,Var3] for recent progress on the topic. ...
This work is devoted to the study of families of infinite parabolic iterated function systems (PIFS) on a closed interval parametrized by with overlaps. We show that the Hausdorff dimension and absolute continuity of ergodic projections through the families of infinite PIFS are decided \emph{a.e.} by the growth rate of the entropy of the sequence of concentrating measures and Lyapunov exponents of the family of truncated PIFS, under transversality of the families essentially. We also give an estimation on the upper bound of the Hausdorff dimension of parameters where the corresponding ergodic projections admit certain dimension drop. The setwise topology on the space of measures enables us to approximate the families of the infinite systems by families of its finite truncated sub-systems, which plays the key role throughout our work.
... Self-similar sets and measures are natural and important objects in fractal geometry, at the interface of geometric measure theory, ergodic theory, number theory and harmonic analysis (see the recent surveys [14,17,32]). Spectacular and influential advances in the dimension theory of these objects have been achieved in the recent period [12,18,15,19,3,33,37,38], in particular in connection with the resolutions of Furstenberg's conjectures on the Hausdorff dimension of the sums and intersections of ×2and ×3-invariant sets on the 1-dimensional torus. ...
Let be a self-similar measure generated by an IFS of similarities on (). When is dimensional regular (see Definition~1.1), we give an explicit formula for the -spectrum of over [0,1], and show that is differentiable over (0,1] and the multifractal formalism holds for at any . We also verify the validity of the multifractal formalism of over for two new classes of overlapping algebraic IFSs by showing that the asymptotically weak separation condition holds. For one of them, the proof appeals to a recent result of Shmerkin on the -spectrum of self-similar measures.
... Questions on selfsimilar sets and measures with overlaps have been considered in many papers (e.g. [3], [10], [26], [27], [31], [32], [34]) and are known to be extremely difficult. ...
We show that for any two homogeneous Cantor sets with sum of Hausdorff dimensions that exceeds 1, one can create an interval in the sumset by applying arbitrary small perturbations (without leaving the class of homogeneous Cantor sets). In our setting the perturbations have more freedom than in the setting of the Palis conjecture, so our result can be viewed as an affirmative answer to a weaker form of the Palis conjecture. We also consider self-similar sets with overlaps on the real line (not necessarily homogeneous) and show that one can create an interval by applying arbitrary small perturbations if the uniform self-similar measure has L 2 L^2 -density.
... The main reason we have decided to return to this topic is a recent breakthrough made by Hochman [9] (see also [2,Theorem 19] for a detailed explanation). Theorem 1.2 (Hochman, 2014). ...
Let β ∈ (1, 2) and let Hβ denote Garsia’s entropy for the Bernoulli convolution μβ associated with β. In the present article we show that Hβ > 0.82 for all β ∈ (1, 2) and improve this bound for certain ranges. Combined with recent results by Hochman and Breuillard-Varjú, this yields (Formula presented.) for all β ∈ (1, 2). In addition, we show that if an algebraic β is such that (Formula presented.) for some k ⩾ 2, then (Formula presented.). Such is, for instance, any root of a Pisot number which is not a Pisot number itself.
... In fact it can be derived from the G case: indeed the action of G factors through that of G, and the fibers of the factor map G → G has two points. Thus if we start with a measure on G and project it to G, then neither the random walk entropy, nor amenability of G µ , is affected; and the results for G may be lifted to G. The proof of Theorem 1.1 is an outgrowth of methods from [16, 17], see also [12], which dealt with the dimension of self-similar measures in Euclidean space. The stationary measure ν is in many respects like a self-similar measure: when µ is finitely supported, stationarity implies that ν it decomposes into " copies " of itself via ν = A∈supp µ µ(A) · Aν. ...
Let be a measure on generating a non-compact and totally irreducible subgroup, let denote its Lyapunov exponent, and let be the associated stationary (Furstenberg) measure for the action on the projective line. We prove that if is supported on finitely many matrices with algebraic entries, then where is the random walk entropy of , and denotes pointwise dimension. In particular, for every , there is a neighborhood U of the identity in such that if a measure is supported on algebraic matrices with all atoms of size at least , and generates a group which is non-compact and totally irreducible, then its stationary measure satisfies .
... The main reason we have decided to return to this topic is a recent breakthrough made by Hochman [8] (see also [2,Theorem 19] for a detailed explanation). ...
Let and let denote Garsia's entropy for the Bernoulli convolution associated with . In the present paper we show that for all and improve this bound for certain ranges. Combined with a recent result by Hochman, this yields for all algebraic . In addition, we show that if an algebraic is such that for some , then . Such is, for instance, any root of a Pisot number which is not a Pisot number itself.
The entropic doubling σent[X] of a random variable X X taking values in an abelian group G G is a variant of the notion of the doubling constant σ[A] of a finite subset A A of G G , but it enjoys somewhat better properties; for instance, it contracts upon applying a homomorphism. In this paper we develop further the theory of entropic doubling and give various applications, including: (1) A new proof of a result of Pálvölgyi and Zhelezov on the “skew dimension” of subsets of ZD with small doubling; (2) A new proof, and an improvement, of a result of the second author on the dimension of subsets of ZD with small doubling; (3) A proof that the Polynomial Freiman–Ruzsa conjecture over F2 implies the (weak) Polynomial Freiman–Ruzsa conjecture over Z.
In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The Bernoulli convolution measure μλ is the probability measure corresponding to the law of the random variableξ=∑k=0∞ξkλk, where ξk are i.i.d. random variables assuming values −1 and 1 with equal probability and 12<λ<1. In particular, for Bernoulli convolutions we give a uniform lower bound dimH(μλ)≥0.96399 for all 12<λ<1.
We generalize a result of Sury [Sur92] and prove that uniform discreteness of cocompact lattices in higher rank semisimple Lie groups (first conjectured by Margulis [Mar91]) is equivalent to a weak form of Lehmer’s conjecture. We include a short survey of related results and conjectures.
We exhibit self-similar sets on the line which are not exponentially separated and do not generate any exact overlaps. Our result shows that the exponential separation, introduced by Hochman in his groundbreaking theorem on the dimension of self-similar sets, is too weak to describe the full theory.
We present a self-contained proof of a formula for the dimensions of self-similar measures on the real line under exponential separation (up to the proof of an inverse theorem for the norm of convolutions). This is a special case of a more general result of the author from Shmerkin (Ann Math, 2019), and one of the goals of this survey is to present the ideas in a simpler, but important, setting. We also review some applications of the main result to the study of Bernoulli convolutions and intersections of self-similar Cantor sets.
By a Cantor-like measure we mean the unique self-similar probability measure satisfying where for integers and probabilities . In the uniform case we show how one can compute the entropy and Hausdorff dimension to arbitrary precision. In the non-uniform case we find bounds on the entropy.
Let β(1,2) be a Pisot number and let Hβ denote Garsia’s entropy for the Bernoulli convolution associated with β. Garsia, in 1963, showed that Hβ<1 for any Pisot β. For the Pisot numbers which satisfy xm=xm−1+xm−2++x+1 (with m≥2), Garsia’s entropy has been evaluated with high precision by Alexander and Zagier for m=2 and later by Grabner, Kirschenhofer and Tichy for m≥3, and it proves to be close to 1. No other numerical values for Hβ are known. In the present paper we show that Hβ>0.81 for all Pisot β, and improve this lower bound for certain ranges of β. Our method is computational in nature.
The paper is devoted to a study of the exit boundary of random walks on discrete groups and related topics. We give an entropic criterion for triviality of the boundary and prove an analogue of Shannon's theorem for entropy, obtain a boundary triviality criterion in terms of the limit behavior of convolutions and prove a conjecture of Furstenberg about existence of a nondegenerate measure with trivial boundary on any amenable group. We directly connect Kesten's and Folner's amenability criteria by consideration of the spectral measure of the Markov transition operator. Finally we give various examples, some of which disprove some old conjectures.
It is shown that the entropy of a sum of independent random vectors is a submodular set function, and upper bounds on the entropy of sums are obtained as a result in both discrete and continuous settings. These inequalities complement the lower bounds provided by the entropy power inequalities of Madiman and Barron (2007). As applications, new inequalities for the determinants of sums of positive-definite matrices are presented.
. An algorithm is given for computing the Hausdorff dimension of the set(s) = (fi ; D) of real numbers with representations x = P 1 n=1 dn fi Gamman , where each dn 2 D, a finite set of "digits", and fi ? 0 is a Pisot number. The Hausdorff dimension is shown to be log = log fi, where is the top eigenvalue of a finite 0-1 matrix A, and a simple algorithm for generating A from the data fi; D is given. 1. Introduction This paper concerns the set(s) = (fi; D) of real numbers with representations x = P 1 n=1 d n fi Gamman , where each d n 2 D, a finite set of "digits", and fi ? 0. These sets have been the subject of several recent studies. Keane, Smorodinsky, and Solomyak [3] considered the special case D = f0; 1; 3g and fi 2 (2:5; 3): they showed that although for almost every fi 2 (2:5; 3) the Hausdorff dimension of is 1, there is a sequence fi k of algebraic integers in (2.5, 3) such that the dimension of is less than 1. Pollicott and Simon [5] showed, more generally, that...
The distribution νλ of the random series random series Σ±λn is the infinite convolution product of These measures have been studied since the 1930’s, revealing connections with harmonic analysis, the theory of algebraic numbers, dynamical systems, and Hausdorff dimension estimation. In this survey we describe some of these connections, and the progress that has been made so far on the fundamental open problem: For which λ∈ is νλ, absolutely continuous?
Our main goal is to present an exposition of results obtained by Erdős, Kahane and the authors on this problem. Several related unsolved problems are collected at the end of the paper.
We show that for any finitely generated group of matrices that is not virtually solvable, there is an integer m such that, given an arbitrary finite generating set for the group, one may find two elements a and b that are both products of at most m generators, such that a and b are free generators of a free subgroup. This uniformity result improves the original statement of the Tits alternative.
We exhibit infinite, solvable, virtually abelian groups with a fixed number of generators, having arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3-generator non-virtually nilpotent polycyclic groups of algebraic entropy tending to zero. All these examples are obtained by taking appropriate quotients of finitely presented groups mapping onto the first Grigorchuk group. Comment: 8 pages; to appear in Archiv der Mathematik
Let k be a local field, and G a linear group over k. We prove that either G contains a relatively open solvable subgroup, or it contains a relatively dense free subgroup. This result has applications in dynamics, Riemannian foliations and profinite groups.
Algebraic geometry affine algebraic groups lie algebras homogeneous spaces chracteristic 0 theory semisimple and unipoten elements solvable groups Borel subgroups centralizers of Tori structure of reductive groups representations and classification of semisimple groups survey of rationality properties.
We prove that Bernoulli convolutions are absolutely continuous provided the parameter lambda is an algebraic number sufficiently close to 1 depending on the Mahler measure of lambda.
Dans cet article, nous introduisons la notion de semi-groupe fortement automatique, qui entraîne la notion d’automaticité des semi-groupes usuelle. On s’intéresse particulièrement aux semi-groupes de développements en base β, pour lesquels on obtient un critère de forte automaticité.
We study a higher dimensional Lehmer problem, or alternatively the Lehmer problem for a power of the multiplicative group. More precisely, if α1, . . . , αn are multiplicatively independent algebraic numbers, we provide a lower bound for the product of the heights of the αi's in terms of the degree D of the number field generated by the αi's. This enables us to study the successive minima for the height function in a given number field. Our bound is a generalisation of an earlier result of Dobrowolski and is best possible up to a power of log(D). This, in particular, shows that the Lehmer problem is true for number fields having a «small» Galois group.
We introduce a conjugation invariant normalized height ĥ(F) on finite subsets of matrices F in GLd(ō{ogonek} and describe its properties. In particular, we prove an analogue of the Lehmer problem for this height by showing that ĥ(F) > ε whenever F generates a nonvirtually solvable subgroup of GLd(ō{ogonek}, where ε = ε(d) > 0 is an absolute constant. This can be seen as a global adelic analog of the classical Margulis Lemma from hyperbolic geometry. As an application we prove a uniform version of the classical Burnside-Schur theorem on torsion linear groups. In a companion paper we will apply these results to prove a strong uniform version of the Tits alternative.
We prove that complex Bernoulli convolutions are absolutely continuous in the
supercritical parameter region, outside of an exceptional set of parameters of
zero Hausdorff dimension. Similar results are also obtained in the biased case,
and for other parametrized families of self-similar sets and measures in the
complex plane, extending earlier results.
Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This second edition of this acclaimed text presents results of both classic and recent matrix analysis using canonical forms as a unifying theme and demonstrates their importance in a variety of applications. This thoroughly revised and updated second edition is a text for a second course on linear algebra and has more than 1,100 problems and exercises, new sections on the singular value and CS decompositions and the Weyr canonical form, expanded treatments of inverse problems and of block matrices, and much more.
We study a higher dimensional Lehmer problem, or alternatively the Lehmer problem for a power of the multiplicative group. More precisely, if alpha(1),..., alpha(n) are multiplicatively independent algebraic numbers, we provide a lower bound for the product of the heights of the alpha(i)'s in terms of the degree D of the number field generated by the alpha(i)'s. This enables us to study the successive minima for the height function in a given number field. Our bound is a generalisation of an earlier result of Dobrowolski and is best possible up to a power of log(D). This, in particular, shows that the Lehmer problem is true for number fields having a < > Galois group.
Additive combinatorics is the theory of counting additive structures in sets. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections with areas such as number theory, ergodic theory and graph theory. This graduate-level 2006 text will allow students and researchers easy entry into this fascinating field. Here, the authors bring together in a self-contained and systematic manner the many different tools and ideas that are used in the modern theory, presenting them in an accessible, coherent, and intuitively clear manner, and providing immediate applications to problems in additive combinatorics. The power of these tools is well demonstrated in the presentation of recent advances such as Szemerédi's theorem on arithmetic progressions, the Kakeya conjecture and Erdos distance problems, and the developing field of sum-product estimates. The text is supplemented by a large number of exercises and new results.
1. The field of values 2. Stable matrices and inertia 3. Singular value inequalities 4. Matrix equations and Kronecker products 5. Hadamard products 6. Matrices and functions 7. Totally positive matrices.
We prove that the set of exceptional such that the
associated Bernoulli convolution is singular has zero Hausdorff dimension, and
likewise for biased Bernoulli convolutions, with the exceptional set
independent of the bias. This improves previous results by Erd\"os, Kahane,
Solomyak, Peres and Schlag, and Hochman. A theorem of this kind is also
obtained for convolutions of homogeneous self-similar measures. The proofs are
very short, and rely on old and new results on the dimensions of self-similar
measures and their convolutions, and the decay of their Fourier transform.
An entropy was introduced by A. Garsia to study certain infinitely convolved Bernoulli measures (ICBMs) μβ, and showed it was strictly less than 1 for β the reciprocal of a Pisot-Vijayarghavan number. However, it is impossible to estimate values from Garsia’s work. The first author and J. A. Yorke have shown this entropy is closely related to the ‘information dimension’ of the attractors of fat baker transformations Tβ. When the entropy is strictly less than 1, the attractor is a type of strange attractor. In this paper, the entropy of μβis estimated for the case when β = ø-1, whereø is the golden ratio. The estimate is fine enough to determine the entropy to several decimal places. The method of proof is totally unlike usual methods for determining dimensions of attractors; rather a relation with the Euclidean algorithm is exploited, and the proof has a number-theoretic flavour. It suggests that some interesting features of the Euclidean algorithm remain to be explored.
In 1878, Jordan showed that a finite subgroup of GL(n, ) must possess an abelian normal subgroup whose index is bounded by a function of n alone. We will give the optimal bound for all n; for n ≥ 71, it is (n + 1)!, afforded by the symmetric group Sn+1. We prove a 'replacement theorem' that enables us to study linear groups by breaking them down into individual primitive constituents and we give detailed information about the structure of the groups that achieve the optimal bounds, for every degree n. Our proof relies on known lower bounds for the degrees of faithful representations of each quasisimple group, depending on the classification of finite simple groups, through the use of the bounds for primitive groups that the author has previously obtained.
We study the Hausdorff dimension of self-similar sets and measures on the
line. We show that if the dimension is smaller than the minimum of 1 and the
similarity dimension, then at small scales there are super-exponentially close
cylinders. This is a step towards the folklore conjecture that such a drop in
dimension is explained only by exact overlaps, and confirms the conjecture in
cases where the contraction parameters are algebraic. It also gives an
affirmative answer to a conjecture of Furstenberg, showing that the projections
of the "1-dimensional Sierpinski gasket" in irrational directions are all of
dimension 1.
As another consequence, if a family of self-similar sets or measures is
parametrized in a real-analytic manner, then, under an extremely mild
non-degeneracy condition, the set of "exceptional" parameters has Hausdorff
dimension 0. Thus, for example, there is at most a zero-dimensional set of
parameters 1/2<r<1 such that the corresponding Bernoulli convolution has
dimension <1, and similarly for Sinai's problem on iterated function systems
that contract on average.
A central ingredient of the proof is an inverse theorem for the growth of
Shannon entropy of convolutions of probability measures. For the dyadic
partition D_n of the line into intervals of length 1/2^n, we show that if
H(nu*mu,D_n)/n < H(mu,D_n)/n + delta for small delta and large n, then, when
restricted to random element of a partition D_i, 0<i<n, either mu is close to
uniform or nu is close to atomic. This should be compared to results in
additive combinatorics that give the global structure of measures satisfying
H(nu*mu,D_n)/n < H(mu,D_n)/n + O(1/n).
A Mandelbrot set for pairs of complex
linear maps was introduced by Barnsley and Harrington
in 1985. Bousch proved that is locally
connected, and Odlyzko and Poonen studied the related set
of all complex roots of polynomials with coefficients 0 and 1.
We give an algorithm to construct this set and study
its geometric structure. In contrast to the Mandelbrot
set for quadratic maps, it is shown that is not simply connected.
In this mostly expository paper, we discuss the strong uniform Tits Alternative and give a complete proof of it in the special case of GL 2 (C). The main arithmetic ingredient, the height gap theorem, is also given a complete treatment in that case. We then prove several applications involving expansion properties of SL d (Z=pZ), a uniform l 2 spectral gap, and diophantine properties of subgroups of GL d (C).
We prove that if f (x) = n−1 k=0 a k x k is a polynomial with no cyclotomic factors whose coefficients satisfy a k ≡ 1 mod 2 for 0 ≤ k < n, then Mahler's measure of f satisfies log M(f) ≥ log 5 4 1 − 1 n . This resolves a problem of D. H. Lehmer [12] for the class of polynomials with odd coefficients. We also prove that if f has odd coefficients, degree n − 1, and at least one noncyclotomic factor, then at least one root α of f satisfies |α| > 1 + log 3 2n , resolving a conjecture of Schinzel and Zassenhaus [21] for this class of poly-nomials. More generally, we solve the problems of Lehmer and Schinzel and Zassenhaus for the class of polynomials where each coefficient satisfies a k ≡ 1 mod m for a fixed integer m ≥ 2. We also characterize the polynomials that appear as the noncyclotomic part of a polynomial whose coefficients satisfy a k ≡ 1 mod p for each k, for a fixed prime p. Last, we prove that the smallest Pisot number whose minimal polynomial has odd coefficients is a limit point, from both sides, of Salem [19] numbers whose minimal polynomials have coef-ficients in {−1, 1}.
Half-title pageSeries pageTitle pageCopyright pageDedicationPrefaceAcknowledgementsContentsList of figuresHalf-title pageIndex
Information theory answers two fundamental questions in communication theory: what is the ultimate data compression (answer: the entropy H), and what is the ultimate transmission rate of communication (answer: the channel capacity C). For this reason some consider information theory to be a subset of communication theory. We will argue that it is much more. Indeed, it has fundamental contributions to make in statistical physics (thermodynamics), computer science (Kolmogorov complexity or algorithmic complexity), statistical inference (Occam's Razor: “The simplest explanation is best”) and to probability and statistics (error rates for optimal hypothesis testing and estimation). The relationship of information theory to other fields is discussed. Information theory intersects physics (statistical mechanics), mathematics (probability theory), electrical engineering (communication theory) and computer science (algorithmic complexity). We describe these areas of intersection in detail.
Let G = ( G , +) be an additive group. The sumset theory of Plünnecke and Ruzsa gives several relations between the size of sumsets A + B of finite sets A , B , and related objects such as iterated sumsets kA and difference sets A − B , while the inverse sumset theory of Freiman, Ruzsa, and others characterizes those finite sets A for which A + A is small. In this paper we establish analogous results in which the finite set A ⊂ G is replaced by a discrete random variable X taking values in G , and the cardinality | A | is replaced by the Shannon entropy H ( X ). In particular, we classify those random variables X which have small doubling in the sense that H ( X 1 + X 2 ) = H ( X ) + O (1) when X 1 , X 2 are independent copies of X , by showing that they factorize as X = U + Z , where U is uniformly distributed on a coset progression of bounded rank, and H ( Z ) = O (1).
When G is torsion-free, we also establish the sharp lower bound \Ent(X+X) \geq \Ent(X) + \frac{1}{2} \log 2 - o(1) , where o (1) goes to zero as H ( X ) → ∞.
Let be an iterated function system (IFS) on with attractor K. Let denote the one-sided full shift over the alphabet . We define the projection entropy function on the space of invariant measures on associated with the coding map , and develop some basic ergodic properties about it. This concept turns out to be crucial in the study of dimensional properties of invariant measures on K. We show that for any conformal IFS (resp., the direct product of finitely many conformal IFS), without any separation condition, the projection of an ergodic measure under is always exactly dimensional and, its Hausdorff dimension can be represented as the ratio of its projection entropy to its Lyapunov exponent (resp., the linear combination of projection entropies associated with several coding maps). Furthermore, for any conformal IFS and certain affine IFS, we prove a variational principle between the Hausdorff dimension of the attractors and that of projections of ergodic measures. Comment: 60 pages
Using a theorem of J. Groves we give a ping-pong proof of Osin's uniform exponential growth for solvable groups. We discuss slow exponential growth and show that this phenomenon disappears as one passes to a finite index subgroup.