58 citations · 373 across the 79 of their papers we have counts for
50 papers · 1 filter
On some results of Korobov and Larcher and Zaremba's conjecture
Ilya D. Shkredov
We prove, in particular, the well--known Zaremba conjecture from the theory of continued fractions for any prime denominator. More precisely, we show, firstly, that under some mild…
On the determinants of matrices with elements from arbitrary sets
Ilya D. Shkredov, Igor E. Shparlinski
Recently there has been several works estimating the number of matrices with elements from some finite sets of arithmetic interest and of a given determina…
On the distribution of quadratic residues
Ilya D. Shkredov
In our paper, we apply additive-combinatorial methods to study the distribution of the set of squares in the prime field. We obtain the best upper bound on the number…
Some applications of representation theory to the sum-product phenomenon
Ilya D. Shkredov
In our paper, we introduce a new method for estimating incidences via representation theory. We obtain several applications to various sums with multiplicative characters and to Za…
The sum-product problem for integers with few prime factors
Brandon Hanson, Misha Rudnev, Ilya Shkredov +1
It was asked by E. Szemerédi if, for a finite set , one can improve estimates for , under the constraint that all integers involved ha…
Some new results on the higher energies I
Ilya D. Shkredov
We obtain a generalization of the recent Kelley--Meka result on sets avoiding arithmetic progressions of length three. In our proof we develop the theory of the higher energies. Al…
On some multiplicative properties of large difference sets
Ilya D. Shkredov
In our paper we study multiplicative properties of difference sets for large sets in the case of composite . We obtain a quantitative…
On Korobov bound concerning Zaremba's conjecture
Nikolay Moshchevitin, Brendan Murphy, Ilya Shkredov
We prove in particular that for any sufficiently large prime there is such that all partial quotients of are bounded by . For composite…
The popularity gap
Vsevolod F. Lev, Ilya D. Shkredov
Suppose that is a finite, nonempty subset of a cyclic group of either infinite or prime order. We show that if the difference set is ``not too large'', then there is a no…
On the random Chowla conjecture
Oleksiy Klurman, Ilya D. Shkredov, Max Wenqiang Xu
We show that for a Steinhaus random multiplicative function and any polynomial of which is not of the form $…
On a paucity result in Incidence Geometry
Ilya D. Shkredov
We obtain some asymptotic formulae (with power savings in their error terms) for the number of quadruples in the Cartesian product of an arbitrary set and fo…
On a girth-free variant of the Bourgain-Gamburd machine
Ilya D. Shkredov
A variant of the Bourgain-Gamburd machine without using any girth bounds is obtained. Also, we find series of applications of %our result the Bourgain-Gamburd machine to problems o…
On an application of higher energies to Sidon sets
Ilya D. Shkredov
We show that for any finite set and an arbitrary there is such that the higher energy is at most un…
Energy bounds for modular roots and their applications
Bryce Kerr, Ilya D. Shkredov, Igor E. Shparlinski +1
We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebra…
Number of solutions in abelian groups and application to counting independent sets in hypergraphs
Aliaksei Semchankau, Dmitry Shabanov, Ilya Shkredov
The paper deals with a problem of Additive Combinatorics. Let be a finite abelian group of order . We prove that the number of subset triples $A,B,C\subset {\mathb…
On some applications of GCD sums to Arithmetic Combinatorics
Ilya D. Shkredov
Using GCD sums, we show that the set of the primes has small common multiplicative energy with an arbitrary exponentially big integer set and, in particular, size of any arithm…
On the distribution of modular square roots of primes
Ilya D. Shkredov, Igor E. Shparlinski, Alexandru Zaharescu
We use recent bounds on bilinear sums with modular square roots to study the distribution of solutions to congruences with primes and integers $q \l…
Character sums estimates and an application to a problem of Balog
Tomasz Schoen, Ilya D. Shkredov
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 conc…
Growth in Chevalley groups relatively to parabolic subgroups and some applications
Ilya D. Shkredov
Given a Chevalley group and a parabolic subgroup , we prove that for any set there is a certain growth of relatively to , namel…
Small doubling in prime-order groups: from to
Vsevolod F. Lev, Ilya D. Shkredov
Improving upon the results of Freiman and Candela-Serra-Spiegel, we show that for a non-empty subset with prime and , (i) if $|A+A|<2.59|A|…
On a modular form of Zaremba's conjecture
Nikolay G. Moshchevitin, Ilya D. Shkredov
We prove that for any prime there is a divisible by number such that for a certain positive integer coprime with the ratio has bounded partial…
Modular hyperbolas and bilinear forms of Kloosterman sums
Ilya D. Shkredov
In this paper we study incidences for hyperbolas in and show how linear sum--product methods work for such curves. As an application we give a purely combinatorial p…
A note on the set
Pierre-Yves Bienvenu, François Hennecart, Ilya Shkredov
Let a large enough prime number. When is a subset of of cardinality , then an application of Cauchy-Davenport Theorem gives…
Popular Products and Continued Fractions
Nikolay Moshchevitin, Brendan Murphy, Ilya Shkredov
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 sha…