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## Publications

Publications (138)

We obtain several lower bounds on the product set of combinatorial cubes, as well as some non–trivial upper estimates for the multiplicative energy of such sets.

For set \(A\subset \mathbb{F}_p^*\) define
by \(\mathsf{sf}(A)\) the size of the largest
sum-free subset of A. Alon and Kleitman [3] showed that
\(\mathsf{sf} (A) \ge |A|/3+O(|A|/p)\).
We prove that if \(\mathsf{sf}(A)-|A|/3\) is small
then the set A must be uniformly distributed on cosets of each large multiplicative subgroup. Our argument relies...

Improving upon the results of Freiman and Candela-Serra-Spiegel, we show that for a non-empty subset A⊆Fp with p prime and |A|<0.0045p, (i) if |A+A|<2.59|A|−3 and |A|>100, then A is contained in an arithmetic progression of size |A+A|−|A|+1, and (ii) if |A−A|<2.6|A|−3, then A is contained in an arithmetic progression of size |A−A|−|A|+1.
The improv...

In this paper we show examples for applications of the Bombieri-Lang conjecture in additive combinatorics, giving bounds on the cardinality of sumsets of squares and higher powers of integers. Using similar methods we give bounds on the sum-product problem for matchings.

We obtain a new bound connecting the first non--trivial eigenvalue of the Laplace operator of a graph and the diameter of the graph, which is effective for graphs with small diameter or for graphs, having the number of maximal paths comparable to the expectation.

We prove new bounds for sums of multiplicative characters over sums of set with small doubling and applying this result we break the square--root barrier in a problem of Balog concerning products of differences in a field of prime order.

For set $A\subset {\mathbb {F}_p}^*$ define by ${\mathsf{sf}}(A)$ the size of the largest sum--free subset of $A.$ Alon and Kleitman showed that ${\mathsf{sf}} (A) \ge |A|/3+O(|A|/p).$ We prove that if ${\mathsf{sf}} (A)-|A|/3$ is small then the set $A$ must be uniformly distributed on cosets of each large multiplicative subgroup. Our argument reli...

Given a Chevalley group ${\mathbf G}(q)$ and a parabolic subgroup $P\subset {\mathbf G}(q)$, we prove that for any set $A$ there is a certain growth of $A$ relatively to $P$, namely, either $AP$ or $PA$ is much larger than $A$. Also, we study a question about intersection of $A^n$ with parabolic subgroups $P$ for large $n$. We apply our method to o...

We show that any set with small Wiener norm has small multiplicative energy. It gives some new bounds for Wiener norm for sets with small product set. Also, we prove that any symmetric subset S of an abelian group has a nonzero Fourier coefficient of size Ω(|S|1∕3).

We prove, for a sufficiently small subset $\mathcal{A}$ of a prime residue field, an estimate on the number of solutions to the equation $(a_1-a_2)(a_3-a_4) = (a_5-a_6)(a_7-a_8)$ with all variables in $\mathcal{A}$. We then derive new bounds on trilinear exponential sums and on the total number of residues equaling the product of two differences of...

We consider in this paper the set of transfer times between two measurable subsets of positive measures in an ergodic probability measure-preserving system of a countable abelian group. If the lower asymptotic density of the transfer times is small, then we prove this set must be either periodic or Sturmian. Our results can be viewed as ergodic-the...

Improving upon the results of Freiman and Candela-Serra-Spiegel, we show that for a non-empty subset $A\subseteq\mathbb F_p$ with $p$ prime and $|A|<0.0045p$, (i) if $|A+A|<2.59|A|-3$ and $|A|>100$, then $A$ is contained in an arithmetic progression of size $|A+A|-|A|+1$, and (ii) if $|A-A|<2.6|A|-3$, then $A$ is contained in an arithmetic progress...

Given a subset of real numbers $A$ with small product $AA$ we obtain a new upper bound for the additive energy of $A$. The proof uses a natural observation that level sets of convolutions of the characteristic function of $A$ have small product with $A$.

We prove that for any prime $p$ there is a divisible by $p$ number $q = O(p^{30})$ such that for a certain positive integer $a$ coprime with $q$ the ratio $a/q$ has bounded partial quotients. In the other direction we show that there is an absolute constant $C>0$ such that for any prime $p$ exist divisible by $p$ number $q = O(p^{C})$ and a number...

We obtain some new results about products of large and small sets in the Heisenberg group as well as in the affine group over the prime field. We apply these growth results to Freiman’s isomorphism in nonabelian groups.

We obtain some new results on products of large and small sets in the Heisenberg group as well as in the affine group over the prime field. Also, we derive an application of these growth results to Freiman's isomorphism in nonabelian groups.

In this paper we study incidences for hyperbolas in $\mathbf{F}_p$ and show how linear sum--product methods work for such curves. As an application we give a purely combinatorial proof of a nontrivial upper bound for bilinear forms of Kloosterman sums.

This note proves that there exists positive constants c 1 and c 2 such that for all finite A⊂R with |A+A|≤|A| 1+c 1 we have |AAA|≫|A| 2+c 2 .

An upper bound for the multiplicative energy of the spectrum of an arbitrary subset of \(\mathbb{F}_{p}\) is obtained. Apparently, at present, this is the best bound.

This paper aims to study in more depth the relation between growth in matrix groups by multiplication and geometric incidence estimates, associated with the sum-product phenomenon. It presents streamlined proofs of Helfgott's theorems on growth in groups $SL (\mathbf{F}_p)$ and $Aff(\mathbf{F}_p)$, avoiding these estimates. In the former case, for...

This note proves that there exists positive constants $c_1$ and $c_2$ such that for all finite $A \subset \mathbb R$ with $|A+A| \leq |A|^{1+c_1}$ we have $|AAA| \gg |A|^{2+c_2}$.

It is established that there exists an absolute constant c > 0 such that for any finite set A of positive real numbers |AA + A| >> |A|3/2+c On the other hand, we give an explicit construction of a finite set A ⊂ R such that |AA + A| = o(|A|²), disproving a conjecture of Balog.

Let $F$ be a field and a finite $A\subset F$ be sufficiently small in terms of the characteristic $p$ of $F$ if $p>0$. We strengthen the "threshold" sum-product inequality $$|AA|^3 |A\pm A|^2 \gg |A|^6\,,\;\;\;\;\mbox{hence} \;\, \;\;|AA|+|A+A|\gg |A|^{1+\frac{1}{5}},$$ due to Roche--Newton, Rudnev and Shkredov, to $$|AA|+|A-A|\gg |A|^{1+\frac{2}{9...

We prove bounds for the popularity of products of sets with weak additive structure, and use these bounds to prove results about continued fractions. Namely, we obtain a nearly sharp upper bound for the cardinality of Zaremba's set modulo $p$.

Let $A \subset \mathbb{F}_p$ of size at most $p^{3/5}$. We show $$|A+A| + |AA| \gtrsim |A|^{6/5 + c},$$ for $c = 4/305$. Our main tools are the cartesian product point--line incidence theorem of Stevens and de Zeeuw and the theory of higher energies developed by the second author.

We obtain an upper bound for the multiplicative energy of the spectrum of an arbitrary set from $\mathbb{F}_p$, which is the best possible up to the results on exponential sums over subgroups.

We apply geometric incidence estimates in positive characteristic to prove the optimal $L^2 \to L^3$ Fourier extension estimate for the paraboloid in the four-dimensional vector space over a prime residue field. In three dimensions, when $-1$ is not a square, we prove an $L^2 \to L^{\frac{32}{9} }$ extension estimate, improving the previously known...

We obtain a new bound on certain double sums of multiplicative characters improving the range of several previous results. This improvement comes from new bounds on the number of collinear triples in finite fields, which is a classical object of study of additive combinatorics.

In this paper we obtain a series of asymptotic formulae in the sum--product phenomena over the prime field $\mathbf{F}_p$. In the proofs we use usual incidence theorems in $\mathbf{F}_p$, as well as the growth result in ${\rm SL}_2 (\mathbf{F}_p)$ due to Helfgott. Here some of our applications: $\bullet~$ a new bound for the number of the solutions...

We prove new results on additive properties of finite sets $A$ with small multiplicative doubling $|AA|\leq M|A|$ in the category of real/complex sets as well as multiplicative subgroups in the prime residue field. The improvements are based on new combinatorial lemmata, which may be of independent interest. Our main results are the inequality $$ |...

We prove that asymptotically almost surely, the random Cayley sum graph over a finite abelian group $G$ has edge density close to the expected one on every induced subgraph of size at least $\log^c |G|$, for any fixed $c > 1$ and $|G|$ large enough.

Using some new observations connected to higher energies, we obtain quantitative lower bounds on $\max\{|AB|, |A+C| \}$ and $\max\{|(A+\alpha)B|, |A+C|\}$, $\alpha \neq 0$ in the regime when the sizes of finite subsets $A,B,C$ of a field differ significantly.

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, as well as generalising a method therein to give a near-optimal bound for a new expander. The main new results are the following bounds, which hold for any finite set $A \subset \mathbb R$: \be...

We prove several quantitative sum-product type estimates over a general field $\mathbb{F}$, in particular the special case $\mathbb{F}=\mathbb{F}_p$. They are unified by the theme of "breaking the $3/2$ threshold", epitomising the previous state of the art. Our progress is partially due to an observation that the recent point-line incidence estimat...

We prove that for an arbitrary $\varepsilon>0$ and any multiplicative subgroup $\Gamma \subseteq \mathbf{F}_p$, $1\ll |\Gamma| \le p^{2/3 -\varepsilon}$ there are no sets $B$, $C \subseteq \mathbf{F}_p$ with $|B|, |C|>1$ such that $\Gamma=B+C$. Also, we obtain that for $1\ll |\Gamma| \le p^{6/7-\varepsilon}$ and any $\xi\neq 0$ there is no a set $B...

We give a new bound on colinear triples in subgroups of prime finite fields and use it to give some new bounds on exponential sums with trinomials.

A new upper bound for the additive energy of the Heilbronn subgroup is found. Several applications to the distribution of Fermat quotients are obtained.

We prove that if $A\subseteq \{ 1,2,\dots, N \}$ does not contain any solution to the equation $x_1+\dots+x_k=y_1+\dots+y_k$ with distinct $x_1,\dots,x_k,y_1,\dots,y_k\in A$, then $|A|\ll {k^{3/2}}N^{1/k}.

We give a lower bound for Wiener norm of characteristic function of subsets A
from Z_p, p is a prime number, in the situation when exp((log p/log log
p)^{1/3}) \le |A| \le p/3.

We improve a recent result of B. Hanson (2015) on multiplicative character sums with expressions of the type $a + b +cd$ and variables $a,b,c,d$ from four distinct sets of a finite field. We also consider similar sums with $a + b(c+d)$. These bounds rely on some recent advances in additive combinatorics.

We prove that for an arbitrary $\kappa < \frac{5}{31}$ any subset of $\mathbf{F}_p$ avoiding $t$ linear equations with three variables has size less than $O(p/t^\kappa)$. We also find several applications to problems about so--called non--averaging sets, number of collinear triples and mixed energies.

We improve previous sum–product estimates in ℝ; namely, we prove the inequality max{|A + A|, |AA|} ≫ |A|4/3+c, where c is any number less than 5/9813. New lower bounds for sums of sets with small product set are found. We also obtain results on the additive and multiplicative energies; in particular, we improve a result of Balog and Wooley.

We prove new exponents for the energy version of the Erd\H{o}s-Szemer\'edi sum-product conjecture, raised by Balog and Wooley. They match the milestone values established earlier for the standard formulation of the question, both for general fields and the special case of real or complex numbers. Further results are obtained about multiplicative en...

Using Stepanov’s method, we obtain an upper bound for the cardinality of the intersection of additive shifts of several multiplicative subgroups of a finite field. The resulting inequality is applied to a question dealing with the additive decomposability of subgroups.

We prove that finite sets of real numbers satisfying $|AA| \leq |A|^{1+\epsilon}$ with sufficiently small $\epsilon > 0$ cannot have small additive bases nor can they be written as a set of sums $B+C$ with $|B|, |C| \geq 2$. The result can be seen as a real analog of the conjecture of S\'ark\"ozy that multiplicative subgroups of finite fields of pr...

A square matrix V is called rigid if every matrix \({V^\prime}\) obtained by altering a small number of entries of V has sufficiently high rank. While random matrices are rigid with high probability, no explicit constructions of rigid matrices are known to date. Obtaining such explicit matrices would have major implications in computational complex...

In the paper we obtain new estimates for binary and ternary sums of multiplicative characters with additive convolutions of characteristic functions of sets, having small additive doubling. In particular, we improve a result of M.-C. Chang. The proof uses Croot-Sisask almost periodicity lemma.

We prove in particular that if A be a compact convex subset of R^n, and B from R^n be an arbitrary compact set then \mu (A-A) \ll \mu(A+B)^2 / (\sqrt{n} \mu (A)), provided that \mu(B)\ge \mu(A).

In the paper we study two characteristics D^+ (A), D^\times (A) of a set A which play important role in recent results concerning sum-product phenomenon. Also we obtain several variants and improvements of the Balog-Wooley decomposition theorem. In particular, we prove that any finite subset of real numbers can be split into two sets with small qua...

We prove, in particular, that for any finite set of real numbers A with |A/A| \ll |A| one has |A-A| > |A|^{5/3 - o(1)}. Also we show that |3A| > |A|^{2-o(1)} in the case.

We improve a previous sum--products estimates in R, namely, we obtain that max{|A+A|,|AA|} \gg |A|^{4/3+c}, where c any number less than 5/9813. New lower bounds for sums of sets with small the product set are found. Also we prove some pure energy sum--products results, improving a result of Balog and Wooley, in particular.

We prove that for any finite set A of real numbers its difference set D:=A-A has large product set and quotient set, namely, |DD|, |D/D| \gg |D|^{1+c}, where c>0 is an absolute constant. Similar result has place in the prime field F_p for sufficiently small D. It gives, in particular, that multiplicative subgroups of the size less p^{4/5-\eps} cann...

Let $F$ be a field of characteristic $p>2$ and $A\subset F$ have sufficiently
small cardinality in terms of $p$. We improve the state of the art of a variety
of sum-product type inequalities. In particular, we prove that $$ |AA|^2|A+A|^3
\gg |A|^6,\qquad |A(A+A)|\gg |A|^{3/2}. $$ We also prove several two-variable
extractor estimates: ${\displaysty...

These notes basically contain a material of two mini--courses which were read
in G\"{o}teborg in April 2015 during the author visit of Chalmers &
G\"{o}teborg universities and in Beijing in November 2015 during
"Chinese--Russian Workshop on Exponential Sums and Sumsets". The article is a
short introduction to a new area of Additive Combinatorics wh...

We prove, in particular, that if A,G are two arbitrary multiplicative
subgroups of the prime field f_p, |G| < p^{3/4} such that the difference A-A is
contained in G then |A| \ll |\G|^{1/3+o(1)}. Also, we obtain that for any eps>0
and a sufficiently large subgroup G with |G| \ll p^{1/2-eps} there is no
representation G as G = A+B, where A is another...

In the paper we prove, in particular, that for any measurable coloring of the
euclidian plane into two colours there is a monochromatic triangle with some
restrictions on the sides. Also we consider similar problems in finite fields
settings.

We prove new general results on sumsets and difference sets for sets of the Szemerédi-Trotter type. This family includes convex sets, sets with small multiplicative doubling, images of sets under convex/concave maps and others.

A finite set of real numbers is called convex if the differences between
consecutive elements form a strictly increasing sequence. We show that, for any
pair of convex sets $A, B\subset\mathbb R$, each of size $n^{1/2}$, the convex
grid $A\times B$ spans at most $O(n^{37/17}\log^{2/17}n)$ unit-area triangles.
This improves the best known upper boun...

We prove that any multiplicative subgroup G of the prime field f_p with |G| <
p^{1/2} satisfies |3G| \gg |G|^2 / \log |G|. Also, we obtain a bound for the
multiplicative energy of any nonzero shift of G, namely E^* (G+x) \ll |G|^2 log
|G|, where x is an arbitrary nonzero residue.

We improve a result of Solymosi on sum-products in R, namely, we prove that
max{|A+A|,|AA|}\gg |A|^{4/3+c}, where c>0 is an absolute constant. New lower
bounds for sums of sets with small product set are found. Previous results are
improved effectively for sets A from R with |AA| \le |A|^{4/3}.

Several structure results of additive combinatorics are considered. Classical and modern problems of combinatorial number theory, higher-order Fourier analysis, inverse theorems for Gowers norms, higher energies, and the relationship between combinatorial and analytic number theory are discussed.

We give a partial answer to a conjecture of A. Balog, concerning the size of
AA+A, where A is a finite subset of real numbers. Also, we prove several new
results on the cardinality of A:A+A, AA+AA and A:A + A:A.

In the paper we obtain some new upper bounds for exponential sums over
multiplicative subgroups G of F^*_p having sizes in the range [p^{c_1},
p^{c_2}], where c_1,c_2 are some absolute constants close to 1/2. As an
application we prove that in symmetric case G is always an additive basis of
order five, provided by |G| > p^{1/2} log^{1/3} p. Also th...

We prove new general results on sumsets of sets having Szemer\'edi--Trotter
type. This family includes convex sets, sets with small multiplicative
doubling, images of sets under convex/concave maps and others.

Let $F=\mathbb F_q$ be a finite field of positive odd characteristic $p$. We
prove a variety of new estimates which can be derived from the theorem that the
number of incidences between $m$ points and $n$ planes in $PG(3,F)$, with
$m,n=O(p^2)$, is $$O\left((mn)^{\frac{3}{4}} + (m+n)k\right),$$ where $k$
denotes the maximum number of collinear point...

In the paper we prove that any sumset or difference set has large E_3 energy.
Also, we give a full description of families of sets having critical relations
between some kind of energies such as E_k, T_k and Gowers norms. In particular,
we give criteria for a set to be a 1) set of the form H+L, where H+H is small
and L has "random structure", 2) se...

We prove some new bounds for the size of the maximal dissociated subset of
structured (having small sumset, large energy and so on) subsets A of an
abelian group.

It is proved that any continuous function f on the unit circle such that the
sequence e^{in f}, n=1,2,... has small Wiener norm \| e^{in f} \|_A = o
(\frac{\log^{1/22} |n|}{(\log \log |n|)^{3/11}}), is linear. Moreover, we get
lower bounds for Wiener norm of characteristic functions of subsets from Z_p in
the case of prime p.

We describe all sets $A \subseteq \F_p$ which represent the quadratic
residues $R \subseteq \F_p$ as $R=A+A$ and $R=A\hat{+} A$. Also, we consider
the case of an approximate equality $R \approx A+A$ and $R \approx A\hat{+} A$
and prove that $A$ has a structure in the situation.

This paper considers various formulations of the sum-product problem. It is
shown that, for a finite set $A\subset{\mathbb{R}}$,
$$|A(A+A)|\gg{|A|^{\frac{3}{2}+\frac{1}{234}}},$$ giving a partial answer to a
conjecture of Balog. In a similar spirit, it is established that
$$|A(A+A+A+A)|\gg{\frac{|A|^2}{\log{|A|}}},$$ a bound which is optimal up to...

In the paper the author proves a new upper bound for Heilbronn’s exponential sum and obtain some applications of his result to a distribution of Fermat quotients.

Using Stepanov's method and some combinatorial observations we prove a new
upper bound for Heilbronn's exponential sum and obtain a series of applications
of our result to distribution of Fermat quotients.

In the paper we develop the method of higher energies. New upper bounds for
the additive energies of convex sets, sets A with small |AA| and |A(A+1)| are
obtained. We prove new structural results, including higher sumsets, and
develop the notion of dual popular difference sets.

We consider functions f(x, y) whose smallness condition for the rectangular norm implies the smallness of the rectangular norm for f(x, x + y). We also study families of functions with a similar property for the higher Gowers norms. The method of proof is based on a transfer principle for sums between special systems of linear equations.

Answering a question of Cochrane and Pinner, we prove that, for any ε > 0, sufficiently large prime number p and an arbitrary multiplicative subgroup R of the field F* p, p ε ≤ |R| ≤ p 2/3-ε, the following holds |R ± R| ≥ |R| 3/2+δ, where δ > 0 depends on ε only. © 2011 Published by Oxford University Press. All rights reserved.

In the paper we find new inequalities involving the intersections $A\cap
(A-x)$ of shifts of some subset $A$ from an abelian group. We apply the
inequalities to obtain new upper bounds for the additive energy of
multiplicative subgroups and convex sets and also a series another results on
the connection of the additive energy and so--called higher...

In the paper we prove a new upper bound for Heilbronn's exponential sum
and obtain some applications of our result to distribution of Fermat
quotients.

Analogues of the Pyatetskii-Shapiro normality criterion for continued fractions and for f-expandings with finite initial tiling are established, improving some results by Moshchevitin and Shkredov obtained in the 2002 paper “On Pyatetskii-Shapiro criterion of normality.”

We study higher moments of convolutions of the characteristic function of a
set, which generalize a classical notion of the additive energy. Such
quantities appear in many problems of additive combinatorics as well as in
number theory. In our investigation we use different approaches including basic
combinatorics, Fourier analysis and eigenvalues m...

We prove, in particular, that if a subset A of {1, 2,..., N} has no
nontrivial solution to the equation x_1+x_2+x_3+x_4+x_5=5y then the cardinality
of A is at most N e^{-c(log N)^{1/7-eps}}, where eps>0 is an arbitrary number,
and c>0 is an absolute constant. In view of the well-known Behrend construction
this estimate is close to best possible.

A set of reals A={a_1,...,a_2} is called convex if a_{i+1} - a_i > a_i -
a_{i-1} for all i. We prove, in particular, that |A-A| \gg |A|^{8/5} \log{-2/5}
|A|.

Generalizing a result of S.V. Konyagin and D.R. Heath--Brown, we prove, in
particular, that for any multiplicative subgroup R of Z/pZ and any nonzero
elements mu_1,...,mu_k the following holds |R \cap (R+mu_1) \cap ... \cap
(R+mu_k)| \ll_k |R|^{1/2+alpha_k}, provided by 1 \ll_k |R| \ll_k p^{1-\beta_k},
where alpha_k, beta_k are some sequences of po...

In a linear space of dimension n over the field F2, we construct a set A of given density such that the Fourier transform of A is large on a large set, and the intersection of A with any subspace of small dimension is small. The results obtained show, in a certain sense, the sharpness of one theorem of J. Bourgain.

In this survey applications of harmonic analysis to combinatorial number theory are considered. Discussion topics include classical problems of additive combinatorics, colouring problems, higher-order Fourier analysis, theorems about sets of large trigonometric sums, results on estimates for trigonometric sums over subgroups, and the connection bet...

Answering a question of T. Cochrane and C. Pinner, we prove that for any
{\epsilon}>0, sufficiently large prime number p and an arbitrary multiplicative
subgroup R of the field Z/pZ, p^{\epsilon} < |R| < p^{2/3-{\epsilon}} the
following holds |R + R|, |R - R| > |R|^{3/2+{\delta}}, where {\delta}>0 depends
on {\epsilon} only.

We show that for any set A in a finite Abelian group G that has at least c vertical bar A vertical bar(3) solutions to a(1) + a(2) = a(3) + a(4), a(j) is an element of A there exist sets A' subset of A and Lambda subset of G, Lambda = {lambda(1), ..., lambda(t)}, t << c(-1) log vertical bar A vertical bar such that A' is contained in {Sigma(t)(j=1)...

In the paper we obtain some new applications of well--known W. Rudin's theorem concerning lacunary series to problems of combinatorial number theory. We generalize a result of M.-C. Chang on L_2 (L)-norm of Fourier coefficients of a set (here L is a dissociated set), and prove a dual version of the theorem. Our main instrument is computing of eigen...