Sum-product phenomena in F_p: a brief introduction

We investigate exponential sums modulo primes whose phase function is a sparse polynomial, with exponents growing with the prime. In particular, such sums model those which appear in the study of the quantum cat map. While they are not amenable to treatment by algebro-geometric methods such as Weil's bounds, Bourgain (2005) gave a nontrivial estimate for these and more general sums. In this work we obtain explicit bounds with reasonable savings over various types of averaging...

Given $d,s \in \mathbb{N}$, a finite set $A \subseteq \mathbb{Z}$ and polynomials $\varphi_1, \dots, \varphi_{s} \in \mathbb{Z}[x]$ such that $1 \leq deg \varphi_i \leq d$ for every $1 \leq i \leq s$, we prove that \[ |A^{(s)}| + |\varphi_1(A) + \dots + \varphi_s(A) | \gg_{s,d} |A|^{\eta_s} , \] for some $\eta_s \gg_{d} \log s / \log \log s$. Moreover if $\varphi_i(0) \neq 0$ for every $1 \leq i \leq s$, then \[ |A^{(s)}| + |\varphi_1(A) \dots \varphi_s(A) | \gg_{s,d} |A|^{...

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.

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.

In this paper we consider estimating the number of solutions to multiplicative equations in finite fields when the variables run through certain sets with high additive structure. In particular, we consider estimating the multiplicative energy of generalized arithmetic progressions in prime fields and of boxes in arbitrary finite fields and obtain sharp bounds in more general scenarios than previously known. Our arguments extend some ideas of Konyagin and Bourgain and Chang i...

We give an improved bound on the famed sum-product estimate in a field of residue class modulo $p$ ($\mathbb{F}_{p}$) by Erd\H{o}s and Szemeredi, and a non-empty set $A \subset \mathbb{F}_{p}$ such that: $$ \max \{|A+A|,|A A|\} \gg \min \left\{\frac{|A|^{15 / 14} \max \left\{1,|A|^{1 / 7} p^{-1 / 14}\right\}}{(\log |A|)^{2 / 7}}, \frac{|A|^{11 / 12} p^{1 / 12}}{(\log |A|)^{1 / 3}}\right\}, $$ and more importantly: $$\max \{|A+A|,|A A|\} \gg \frac{|A|^{15 / 14}}{(\log |A|)^{2 ...

Let $\mathbb{F}_q$ denote the finite field with $q$ elements where $q=p^l$ is a prime power. Using Fourier analytic tools with a third moment method, we obtain sum-product type estimates for subsets of $\mathbb{F}_q$. In particular, we prove that if $A\subset \mathbb{F}_q$, then $$|AA+A|,|A(A+A)|\gg\min\left\{q, \frac{|A|^2}{q^{\frac{1}{2}}} \right\},$$ so that if $A\ge q^{\frac{3}{4}}$, then $|AA+A|,|A(A+A)|\gg q$.

The main objective of this article is to study the exponential sums associated to Fourier coefficients of modular forms supported at numbers having a fixed set of prime factors. This is achieved by establishing an improvement on Shparlinski's bound for exponential sums attached to certain recurrence sequences over finite fields.

In this paper we start to investigate a new body of questions in additive combinatorics. The fundamental Cauchy--Davenport theorem gives a lower bound on the size of a sumset A+B for subsets of the cyclic group Zp of order p (p prime), and this is just one example of a large family of results. Our aim in this paper is to investigate what happens if we restrict the number of elements of one set that we may use to form the sums. Here is the question we set out to answer: given ...

Let $\beta,\epsilon \in (0,1]$, and $k \geq \exp(122 \max\{1/\beta,1/\epsilon\})$. We prove that if $A,B$ are subsets of a prime field $\mathbb{Z}_{p}$, and $|B| \geq p^{\beta}$, then there exists a sum of the form $$S = a_{1}B \pm \ldots \pm a_{k}B, \qquad a_{1},\ldots,a_{k} \in A,$$ with $|S| \geq 2^{-12}p^{-\epsilon}\min\{|A||B|,p\}$. As a corollary, we obtain an elementary proof of the following sum-product estimate. For every $\alpha < 1$ and $\beta,\delta > 0$, there ...