March 26, 2015
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October 29, 2006
Let A and B be two finite subsets of a field F. In this paper we provide a nontrivial lower bound for |{a+b: a in A, b in B, and P(a,b) not=0}| where $P(x,y)\in F[x,y]$.
July 23, 2011
Assume that $A\subseteq \Fp, B\subseteq \Fp^{*}$, $\1/4\leqslant\frac{|B|}{|A|},$ $|A|=p^{\alpha}, |B|=p^{\beta}$. We will prove that for $p\geqslant p_0(\beta)$ one has $$\sum_{b\in B}E_{+}(A, bA)\leqslant 15 p^{-\frac{\min\{\beta, 1-\alpha\}}{308}}|A|^3|B|.$$ Here $E_{+}(A, bA)$ is an additive energy between subset $A$ and it's multiplicative shift $bA$. This improves previously known estimates of this type.
March 17, 2021
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 {\it Sali{\'e}} sums and to a new equidistribution estimate for the set of modular roots of primes.
August 18, 2011
Recently, additive combinatorics has blossomed into a vibrant area in mathematical sciences. But it seems to be a difficult area to define - perhaps because of a blend of ideas and techniques from several seemingly unrelated contexts which are used there. One might say that additive combinatorics is a branch of mathematics concerning the study of combinatorial properties of algebraic objects, for instance, Abelian groups, rings, or fields. This emerging field has seen tremend...
August 30, 2017
We consider the problem of identity testing and recovering (that is, interpolating) of a "hidden" monic polynomials $f$, given an oracle access to $f(x)^e$ for $x\in\mathbb F_q$, where $\mathbb F_q$ is the finite field of $q$ elements and an extension fields access is not permitted. The naive interpolation algorithm needs $de+1$ queries, where $d =\max\{{\rm deg}\ f, {\rm deg }\ g\}$ and thus requires $ de<q$. For a prime $q = p$, we design an algorithm that is asymptotical...
June 19, 2018
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.
October 25, 2012
Let $G_1,..., G_n \in \Fp[X_1,...,X_m]$ be $n$ polynomials in $m$ variables over the finite field $\Fp$ of $p$ elements. A result of {\'E}. Fouvry and N. M. Katz shows that under some natural condition, for any fixed $\varepsilon$ and sufficiently large prime $p$ the vectors of fractional parts $$ (\{\frac{G_1(\vec{x})}{p}},...,\{\frac{G_n(\vec{x})}{p}}), \qquad \vec{x} \in \Gamma, $$ are uniformly distributed in the unit cube $[0,1]^n$ for any cube $\Gamma \in [0, p-1]^m$ wi...
April 19, 2007
For a prime $p$ and an absolutely irreducible modulo $p$ polynomial $f(U,V) \in \Z[U,V]$ we obtain an asymptotic formulas for the number of solutions to the congruence $f(x,y) \equiv a \pmod p$ in positive integers $x \le X$, $y \le Y$, with the additional condition $\gcd(x,y)=1$. Such solutions have a natural interpretation as solutions which are visible from the origin. These formulas are derived on average over $a$ for a fixed prime $p$, and also on average over $p$ for a ...
November 18, 2023
The well-known $|supp(f)||supp(\widehat{f}|\geq |G|$ inequality gives lower estimation of each supports. In the present paper we give upper estimation under arithmetic constrains. The main notion will be the additive energy which plays a central role in additive combinatorics. We prove an uncertainty inequality that shows a trade-off between the total changes of the indicator function of a subset $A\subseteq \mathbb F^n_2$ and the additive energy of $A$ and the Fourier spectr...
January 12, 2010
Let $P = A\times A \subset \mathbb{F}_p \times \mathbb{F}_p$, $p$ a prime. Assume that $P= A\times A$ has $n$ elements, $n<p$. See $P$ as a set of points in the plane over $\mathbb{F}_p$. We show that the pairs of points in $P$ determine $\geq c n^{1 + {1/267}}$ lines, where $c$ is an absolute constant. We derive from this an incidence theorem: the number of incidences between a set of $n$ points and a set of $n$ lines in the projective plane over $\F_p$ ($n<\sqrt{p}$) is b...