Péter P. Varjú’s research while affiliated with University of Cambridge and other places

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Publications (35)

Counting rationals and diophantine approximation in missing-digit Cantor sets
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  • File available

March 2024

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75 Reads

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Péter P Varjú

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We establish a new upper bound for the number of rationals up to a given height in a missing-digit set, making progress towards a conjecture of Broderick, Fishman, and Reich. This enables us to make novel progress towards another conjecture of those authors about the corresponding intrinsic diophantine approximation problem. Moreover, we make further progress towards conjectures of Bugeaud-Durand and Levesley-Salp-Velani on the distribution of diophantine exponents in missing-digit sets.

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Exponential multiple mixing for commuting automorphisms of a nilmanifold

October 2023

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7 Reads

Ergodic Theory and Dynamical Systems

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PÉTER P. VARJÚ

Let l∈N≥1l\in \mathbb {N}_{\ge 1} and α:Zl→Aut(N)\alpha : \mathbb {Z}^l\rightarrow \text {Aut}(\mathscr {N}) be an action of Zl\mathbb {Z}^l by automorphisms on a compact nilmanifold N\mathscr{N} . We assume the action of every α(z)\alpha (z) is ergodic for z∈Zl∖{0}z\in \mathbb {Z}^l\smallsetminus \{0\} and show that α\alpha satisfies exponential n -mixing for any integer n≥2n\geq 2 . This extends the results of Gorodnik and Spatzier [Mixing properties of commuting nilmanifold automorphisms. Acta Math. 215 (2015), 127–159].

Cut-off phenomenon for the ax+b Markov chain over a finite field

September 2022

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26 Reads

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14 Citations

Probability Theory and Related Fields

Emmanuel Breuillard

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Péter P. Varjú

We study the Markov chain xn+1=axn+bnx_{n+1}=ax_n+b_n on a finite field Fp{\mathbb {F}}_p, where a∈Fp×a \in {\mathbb {F}}_p^{\times } is fixed and bnb_n are independent and identically distributed random variables in Fp{\mathbb {F}}_p. 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 a∈Fp×a \in {\mathbb {F}}_p^\times . We also obtain weaker, but unconditional, upper bounds for the mixing time.

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On the Multiplicative Group Generated by Two Primes in Z∕QZ

May 2022

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5 Reads

Péter P. Varjú

We study the action of the multiplicative group generated by two prime numbers in Z∕Q Z. More specifically, we study returns to the set ([−Qε, Qε] ∩Z)∕Q Z. This is intimately related to the problem of bounding the greatest common divisor of S-unit differences, which we revisit. Our main tool is the S-adic subspace theorem.

Exponential multiple mixing for commuting automorphisms of a nilmanifold

January 2022

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14 Reads

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Péter P. Varjú

Let l∈N≥1l\in \mathbb{N}_{\geq 1} and α:Zl→Aut(N)\alpha : \mathbb{Z}^l\rightarrow \text{Aut}(\mathscr{N}) be an action of Zl\mathbb{Z}^l by automorphisms on a compact nilmanifold N\mathscr{N}. We assume the action of every α(z)\alpha(z) is ergodic for z∈Zl∖{0}z\in \mathbb{Z}^l\smallsetminus\{0\} and show that α\alpha satisfies exponential n-mixing for any integer n≥2n\geq 2. This extends results of Gorodnik and Spatzier [Acta Math., 215 (2015)].

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Self-similar sets and measures on the line

September 2021

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42 Reads

Péter P. Varjú

We discuss the problem of determining the dimension of self-similar sets and measures on R\mathbf{R}. 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.

On the multiplicative group generated by 2 and 3 in $\mathbf{Z}/Q\mathbf{Z}

April 2021

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5 Reads

Péter P. Varjú

We study the action of the multiplicative group generated by two prime numbers in Z/QZ\mathbf{Z}/Q\mathbf{Z}. More specifically, we study returns to the set ([−Qε,Qε]∩Z)/QZ([-Q^\varepsilon,Q^\varepsilon]\cap \mathbf{Z})/Q\mathbf{Z}. This is intimately related to the problem of bounding the greatest common divisor of S-unit differences, which we revisit. Our main tool is the S-adic subspace theorem.

Mixing time of the Chung–Diaconis–Graham random process

February 2021

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11 Reads

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26 Citations

Probability Theory and Related Fields

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Péter P. Varjú

Define (Xn) on Z/qZ by Xn+1=2Xn+bn, where the steps bn are chosen independently at random from -1,0,+1. The mixing time of this random walk is known to be at most 1.02log2q for almost all odd q (Chung, Diaconis, Graham in Ann Probab 15(3):1148–1165, 1987), and at least 1.004log2q (Hildebrand in Proc Am Math Soc 137(4):1479–1487, 2009). We identify a constant c=1.01136⋯ such that the mixing time is (c+o(1))log2q for almost all odd q. In general, the mixing time of the Markov chain Xn+1=aXn+bn modulo q, where a is a fixed positive integer and the steps bn are i.i.d. with some given distribution in Z, is related to the entropy of a corresponding self-similar Cantor-like measure (such as a Bernoulli convolution). We estimate the mixing time up to a 1+o(1) factor whenever the entropy exceeds (loga)/2.

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Citations (18)

... We finish this discussion on iterated function systems, and in particular on overlapping iterated functions systems, by emphasising that the study of these objects is currently a very active area of research. We refer the reader to the articles [21,22,34,35,39,40] for recent advances in the study of overlapping iterated function systems and their associated self-affine sets. ...

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Quantitative recurrence and the shrinking target problem for overlapping iterated function systems
Self-similar sets and measures on the line
  • Citing Chapter

  • December 2023

Péter P. Varjú

... It will illustrate that the random mapping technique can be effective in constructing strong stationary times in situations where they are difficult to find and have lead to numerous mistakes in the past. While there is room for improvement in our estimates, we hope this new approach will help the understanding of the convergence to equilibrium of related random walks, see for instance Hermon and Thomas [11], Breuillard and Varjú [2], Eberhard and Varjú [9] or Chatterjee and Diaconis [5] for very recent progress in this direction. the matrices of the form where x k,l Z M for 1 ¤ k l ¤ N , the group operation corresponds to the matrix multiplication. ...

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Strong Stationary Times for Finite Heisenberg Walks
Cut-off phenomenon for the ax+b Markov chain over a finite field

Probability Theory and Related Fields

Emmanuel Breuillard

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Péter P. Varjú

... ) We end this introduction with a few remarks about the implications Theorem 2 has on the Fourier decay problem for fractal measures. Recall that a probability measure ν on R is called a Rajchman measure if lim |q|→∞ ν(q) = 0. Combining recent breakthroughs on the Fourier decay problem [13,25,20,4,3,1,5] we know that if a C ω (R) self-conformal IFS admits a non-Rajchman self-conformal measure then: The IFS must be self-similar; It must have contraction ratios that are all powers of some r ∈ (0, 1) such that r −1 is a Pisot number; And, up to affine conjugation, all translates are in Q(r ). Now, it is natural to ask about analogues of this result in C r (R)-regularity, 1 ≤ r ≤ ∞ -see e.g. ...

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How linear can a non-linear hyperbolic IFS be?
Fourier decay of self-similar measures and self-similar sets of uniqueness
  • Citing Article

  • June 2022

Analysis and Partial Differential Equations

Péter P. Varjú

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... In this work we establish cutoff for random walks on randomly generated graphs at an entropic time. There have been multiple recent works proving cutoff at an entropic time, including [4,6,8,9,11,12,15,17,18]. For a more detailed overview please refer to [18]. ...

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Phase transition for random walks on graphs with added weighted random matching
Mixing time of the Chung–Diaconis–Graham random process

Probability Theory and Related Fields

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Péter P. Varjú

... Here, we always condition on ξ n = 0. In this context, [BV19] proved that under the General Riemann Hypothesis (GRH), the polynomial is irreducible a.a.s. Unconditional results [BSK20,BSKK23] show that the probability is bounded away from zero in full generality and under mild restrictions tend to 1 (e.g., ξ i are uniformly distributed on an interval of length ≥ 35). ...

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Irreducibility of Polynomials with Square Coefficients over Finite Fields
Irreducibility of random polynomials of large degree
  • Citing Article

  • January 2019

Acta Mathematica

Emmanuel Breuillard

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Péter P. Varjú

... Let us finally mention the result of [27], which shows that the expansion of a simple random walk on a semi-direct product (Z pZ) d ⋊ SL d (Z pZ), p prime, and that of the projection to SL d (Z pZ), are of the same order. Hence in this case the mixing properties of an affine random walk are essentially determined by the product of random matrices. ...

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Accelerating abelian random walks with hyperbolic dynamics
Spectral gap in the group of affine transformations over prime fields
  • Citing Article

  • November 2016

Annales de la faculté des sciences de Toulouse Mathématiques

Elon Lindenstrauss

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Péter P. Varjú

... The first examples of IFSs with this 'super-exponential concentration' property were given in [Bak;BK3], and there has been further work on the dimension theory of such IFSs [Rap; RV]. There are also longstanding open problems about dimensions and absolute continuity of overlapping self-similar measures such as Bernoulli convolutions [BV;Erd1;Erd2;Var1;Var2;Var3]. ...

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Interpolating between Hausdorff and box dimension
On the dimension of Bernoulli convolutions
  • Citing Article

  • October 2016

The Annals of Probability

Emmanuel Breuillard

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Péter P. Varjú

... The study of Bernoulli convolutions has a long history, going back to Jessen and Wintner [30] that proved that the distribution ν c is either singular or absolutely continuous with respect to the Lebesgue measure (see e.g. Varjú [44] or Peres et al. [38] for recent surveys of the topic and Benaim and El Karoui [8], chapter 1, for a basic introduction). In our setting, the following conclusions can be drawn from the existing literature: ...

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Opinion dynamics in communities with major influencers and implicit social influence via mean-field approximation
Recent progress on Bernoulli convolutions
  • Citing Article

  • August 2016

Péter P. Varjú

... Since the B n 's are bounded and 1 − µ ∈ (0, 1), the partial sums µ n k=0 (1 − µ) k B k converge almost surely, thus Z ∞ is well defined. Moreover, if B is an independent copy of the B n 's, then We remark that in this case, the infinite (random) series n (1 − µ) n B n = Z ∞ /µ is termed the Bernoulli convolution, which has been studied extensively [18,34,38,39,41]. The following characterization of ρ ∞ is well known in this literature; here we provide a different proof using the PDE (3.4): ...

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Fractal opinions among interacting agents
Absolute continuity of Bernoulli convolutions for algebraic parameters
  • Citing Article

  • January 2016

Journal of the American Mathematical Society

Péter P. Varjú

... 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). ...

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Dimension of Bernoulli Convolutions in R^d
Entropy of Bernoulli Convolutions and Uniform Exponential Growth for Linear Groups
  • Citing Article

  • October 2015

Journal d Analyse Mathématique

Emmanuel Breuillard

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Péter P. Varjú