arXiv subjects
Publications and source records attributed to Ilya D. Shkredov.
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, algebraic number theory and the geometry of numbers. We give applications of these results to new bounds on correlations between Salié sums and to a new equidistribution estimate for the set of modular roots of primes.
Let $R=\mathbb{Z}/(2^n-1)\mathbb{Z}$, where $n\geq 3$, and let $S_w\subseteq R$ be the residues whose canonical $n$-digit binary expansion has Hamming weight $w$. We obtain, in particular, an asymptotic formula for the additive energy of $S_w$ \[ E(S_w)=\frac{\left|S_w\right|^4}{|R|}+ \mathcal{O}\left(|R|^3 n^{-3} \right), \] which holds uniformly in $w$. The error term is optimal in order, with a matching lower bound for $w=\lfloor n/2+\sqrt{n} \rfloor$. It follows that triple sums of arbitrary unit dilates have asymptotically uniform representation counts when $\prod_{j=1}^{3} \left|S_{w_j} \right| /\left(|R| n^{-3/5}\right)^3\to\infty$, and that double sums have asymptotically full support when $\left|S_{w_1}\right| \left|S_{w_2} \right|/\left(|R| n^{-3/4}\right)^2\to\infty$. In the proof, modular collisions are represented using a cyclic binary carry automaton; this appears to be a novel approach in this area of problems.
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 conditions, for any sufficiently large $q$, there exists $a$ coprime to $q$ such that all partial quotients of $a/q$ are bounded by $O(\sqrt{\log q})$, and, moreover we find asymptotically tight lower bound for the number of such $a$. Secondly, we obtain a good lower bound for the number $a$ such that the sum of all partial quotients of $a/q$ is bounded by $O(\log q \cdot \sqrt{\log \log q})$. This, accordingly, improves on some results of Korobov and Larcher. Finally, we show that for all sufficiently large $\mathcal{M}$ there are $Ω(q^{1-O(1/\mathcal{M})})$ numbers $a$ coprime to $q$ such that all partial quotients of $a/q$ are bounded by $\mathcal{M}$.
We continue to study the relationship between the size of the sum of a set and the common energy of its subsets. We find a rather sharp subexponential dependence between the doubling constant of a set $A$ and the minimal common energy taken over all partitions of $A$ into two disjoint subsets. As an application, we give a proof of the well--known arithmetic regularity lemma with better dependence on parameters.
Let $A$ be a subset of a finite abelian group such that $A$ has a small difference set $A-A$ and the density of $A$ is small. We prove that, counter--intuitively, the smallness (in terms of $|A-A|$) of the Fourier coefficients of $A$ guarantees that $A$ is correlated with a large Bohr set. Our bounds on the size and the dimension of the resulting Bohr set are close to exact.
We obtain a polynomial criterion for a set to have a small doubling in terms of the common energy of its subsets.
Recently there has been several works estimating the number of $n\times n$ matrices with elements from some finite sets $\mathcal X$ of arithmetic interest and of a given determinant. Typically such results are compared with the trivial upper bound $O(X^{n^2-1})$, where $X$ is the cardinality of $\mathcal X$. Here we show that even for arbitrary sets $\mathcal X\subseteq \mathbb R$,some recent results from additive combinatorics enable us to obtain a stronger bound with a power saving.
The paper obtains the optimal form of the uncertainty principle in the special case of convolution of sets.
We study the concept of universal sets from the additive--combinatorial point of view. Among other results we obtain some applications of this type of uniformity to sets avoiding solutions to linear equations, and get an optimal upper bound for the covering number of general sumsets.
In our paper, we apply additive-combinatorial methods to study the distribution of the set of squares $\mathcal{R}$ in the prime field. We obtain the best upper bound on the number of gaps in $\mathcal{R}$ at the moment and generalize this result for sets with small doubling.
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 Zaremba's conjecture from number theory.
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. Also, we discuss the case of longer arithmetic progressions, as well as a general family of norms, which includes the higher energies norms and Gowers norms.
In our paper we study multiplicative properties of difference sets $A-A$ for large sets $A \subseteq \mathbb{Z}/q\mathbb{Z}$ in the case of composite $q$. We obtain a quantitative version of a result of A. Fish about the structure of the product sets $(A-A)(A-A)$. Also, we show that the multiplicative covering number of any difference set is always small.
Suppose that $A$ is a finite, nonempty subset of a cyclic group of either infinite or prime order. We show that if the difference set $A-A$ is ``not too large'', then there is a nonzero group element with at least as many as $(2+o(1))|A|^2/|A-A|$ representations as a difference of two elements of $A$; that is, the second largest number of representations is, essentially, twice the average. Here the coefficient $2$ is the best possible. We also prove continuous and multidimensional versions of this result, and obtain similar results for sufficiently dense subsets of an arbitrary abelian group.
We develop the theory of the additive dimension ${\rm dim} (A)$, i.e. the size of a maximal dissociated subset of a set $A$. It was shown that the additive dimension is closely connected with the growth of higher sumsets $nA$ of our set $A$. We apply this approach to demonstrate that for any small multiplicative subgroup $Γ$ the sequence $|nΓ|$ grows very fast. Also, we obtain a series of applications to the sum--product phenomenon and to the Balog--Wooley decomposition--type results.
We show that for a Steinhaus random multiplicative function $f:\mathbb{N}\to\mathbb{D}$ and any polynomial $P(x)\in\mathbb{Z}[x]$ of $\text{deg}\ P\ge 2$ which is not of the form $w(x+c)^{d}$ for some $w\in \mathbb{Z}$, $c\in \mathbb{Q}$, we have \[\frac{1}{\sqrt{x}}\sum_{n\le x} f(P(n)) \xrightarrow{d} \mathcal{CN}(0,1),\] where $\mathcal{CN}(0,1)$ is the standard complex Gaussian distribution with mean $0$ and variance $1.$ This confirms a conjecture of Najnudel in a strong form. We further show that there almost surely exist arbitrary large values of $x\ge 1,$ such that $$|\sum_{n\le x} f(P(n))| \gg_{\text{deg}\ P} \sqrt{x} (\log \log x)^{1/2},$$ for any polynomial $P(x)\in\mathbb{Z}[x]$ with $\text{deg}\ P\ge 2,$ which is not a product of linear factors (over $\mathbb{Q}$). This matches the bound predicted by the law of the iterated logarithm. Both of these results are in contrast with the well-known case of linear phase $P(n)=n,$ where the partial sums are known to behave in a non-Gaussian fashion and the corresponding sharp fluctuations are speculated to be $O(\sqrt{x}(\log \log x)^{\frac{1}{4}+\varepsilon})$ for any $\varepsilon>0$.
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 $A \subset \mathbf{R}$ and for the number of quintuplets in $A\times A$ for any subset $A$ of the prime field $\mathbf{F}_p$. Also, we obtain some applications of our results to incidence problems in $\mathbf{F}_p$.
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 of Additive Combinatorics, Number Theory and Probability.