arXiv subjects
Publications and source records attributed to Ilya D. Shkredov.
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 $A \subset \mathbb R$ with $|A+A| \leq |A|^{1+c_1}$ we have $|AAA| \gg |A|^{2+c_2}$.
This paper aims to study in more depth the relation between growth in matrix groups ${\rm SL_2}(\mathbf{F})$ and ${\rm Aff}(\mathbf{F})$ over a field $\mathbf{F}$ by multiplication and geometric incidence estimates, associated with the sum-product phenomenon over $\mathbf{F}$. It presents streamlined proofs of Helfgott's theorems on growth in the $\mathbf{F}_p$-case, which avoid sum-product estimates. For ${\rm SL_2}(\mathbf{F}_p)$, for sets exceeding in size some absolute constant, we improve the lower bound $\frac{1}{1512}$ for the growth exponent, due to Kowalski, to $\frac{1}{21}.$ For the affine group we fetch a sharp theorem of Szőnyi on the number of directions, determined by a point set in $\mathbf{F}_p^2$. We then focus on ${\rm Aff}(\mathbf{F})$ and present a new incidence bound between a set of points and a set of lines in $\mathbf{F}^2$, which explicitly depends on the energy of the set of lines as affine transformations under composition. This bound, strong when the number of lines is considerably smaller than the number of points, yields generalizations of structural theorems of Elekes and Murphy on rich lines in grids. In the special case when the set of lines is also a grid -- relating back to sum-products -- we use growth in ${\rm Aff}(\mathbf{R})$ to obtain a subthreshold estimate on the energy of the set of lines. This yields a unified way to break the ice in various threshold sum-product type energy inequalities. We show this in applications to energy estimates, corresponding to sets $A(A+ A)$, $A+AA$ (also embracing asymmetric versions) as well as $A+B$ when $A$ has small multiplicative doubling and $\sqrt{|A|} \le |B|\le|A|^{1+o(1)}$.
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 exponent $\frac{68}{19}.$
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 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 to the equation $(a_1-a_2) (a_3-a_4) = (a'_1-a'_2) (a'_3-a'_4)$, $\,a_i, a'_i\in A$, $A$ is an arbitrary subset of $\mathbf{F}_p$, $\bullet~$ a new effective bound for multilinear exponential sums of Bourgain, $\bullet~$ an asymptotic analogue of the Balog--Wooley decomposition theorem, $\bullet~$ growth of $p_1(b) + 1/(a+p_2 (b))$, where $a,b$ runs over two subsets of $\mathbf{F}_p$, $p_1,p_2 \in \mathbf{F}_p [x]$ are two non--constant polynomials, $\bullet~$ new bounds for some exponential sums with multiplicative and additive characters.
It is established that there exists an absolute constant $c>0$ such that for any finite set $A$ of positive real numbers $$|AA+A| \gg |A|^{\frac{3}{2}+c}.$$ On the other hand, we give an explicit construction of a finite set $A \subset \mathbb R$ such that $|AA+A|=o(|A|^2)$, disproving a conjecture of Balog.
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 $$ |A-A|^3|AA|^5 \gtrsim |A|^{10}, $$ over the reals, "redistributing" the exponents in the textbook Elekes sum-product inequality and the new best known additive energy bound $\mathsf E(A)\lesssim_M |A|^{49/20}$, which aligns, in a sense to be discussed, with the best known sum set bound $|A+A|\gtrsim_M |A|^{8/5}$. These bounds, with $M=1$, also apply to multiplicative subgroups of $\mathbb F^\times_p$, whose order is $O(\sqrt{p})$. We adapt the above energy bound to larger subgroups and obtain new bounds on gaps between elements in cosets of subgroups of order $Ω(\sqrt{p})$.
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.
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.
We prove new exponents for the energy version of the Erdős-Szemerédi sum-product conjecture, raised by Balog and Wooley. They match the previously established milestone values for the standard formulation of the question, both for general fields and the special case of real or complex numbers, and appear to be the best ones attainable within the currently available technology. Further results are obtained about multiplicative energies of additive shifts and a strengthened energy version of the "few sums, many products" inequality of Elekes and Ruzsa. The latter inequality enables us to obtain a minor improvement of the state-of the art sum-product exponent over the reals due to Konyagin and the second author, up to $\frac{4}{3}+\frac{1}{1509}$. An application of energy estimates to an instance of arithmetic growth in prime residue fields is presented.
Using some new observations connected to higher energies, we obtain quantitative lower bounds on $\max\{|AB|, |A+C| \}$ and $\max\{|(A+α)B|, |A+C|\}$, $α\neq 0$ in the regime when the sizes of finite subsets $A,B,C$ of a field differ significantly.
For a given subset $A\subseteq \mathbb F_q^*$, we study the problem of finding a large packing set $B$ of $A$, that is, a set $B \subseteq \mathbb F_q^*$ such that $|AB|=|A||B|$. We prove the existence of such a $B$ of size $|B|\ge (q-1)/|A/A|$ and show that this bound is in general optimal. The case that $q=p$ is a prime and $A=\{1,2,\ldots,λ\}$ for some positive integer $λ$ is particularly interesting in view of the construction of limited-magnitude error correcting codes. Here we construct a packing set $B$ of size $|B|\gg p (λ\log p)^{-1}$ for any $λ\le c p^{1/2}$ for some explicitly calcuable constant $c$. This result is optimal up to the logarithmic factor.
We prove that for an arbitrary $\varepsilon>0$ and any multiplicative subgroup $Γ\subseteq \mathbf{F}_p$, $1\ll |Γ| \le p^{2/3 -\varepsilon}$ there are no sets $B$, $C \subseteq \mathbf{F}_p$ with $|B|, |C|>1$ such that $Γ=B+C$. Also, we obtain that for $1\ll |Γ| \le p^{6/7-\varepsilon}$ and any $ξ\neq 0$ there is no a set $B$ such that $ξΓ+1=B/B$.
We prove that finite sets of real numbers satisfying $|AA| \leq |A|^{1+ε}$ with sufficiently small $ε> 0$ 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 Sárközy that multiplicative subgroups of finite fields of prime order are additively irreducible.
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 prove that for an arbitrary $κ\le \frac{1}{3}$ any subset of $\mathbf{F}_p$ avoiding $t$ linear equations with three variables has size less than $O(p/t^κ)$. We also find several applications to problems about so--called non--averaging sets, number of collinear triples and mixed energies.