Sum-product phenomena in F_p: a brief introduction

The \emph{sum-product phenomenon} predicts that a finite set $A$ in a ring $R$ should have either a large sumset $A+A$ or large product set $A \cdot A$ unless it is in some sense "close" to a finite subring of $R$. This phenomenon has been analysed intensively for various specific rings, notably the reals $\R$ and cyclic groups $\Z/q\Z$. In this paper we consider the problem in arbitrary rings $R$, which need not be commutative or contain a multiplicative identity. We obtain ...

We improve the exponent in the finite field sum-product problem from $11/9$ to $5/4$, improving the results of Rudnev, Shakan and Shkredov. That is, we show that if $A\subset \mathbb{F}_p$ has cardinality $|A|\ll p^{1/2}$ then \[ \max\{|A\pm A|,|AA|\} \gtrsim |A|^\frac54 \] and \[ \max\{|A\pm A|,|A/A|\}\gtrsim |A|^\frac54\,. \]

Let F_q be the finite field of q elements. Let H be a multiplicative subgroup of F_q^*. For a positive integer k and element b\in F_q, we give a sharp estimate for the number of k-element subsets of H which sum to b.

The main result of this paper is the following: for all $b \in \mathbb Z$ there exists $k=k(b)$ such that \[ \max \{ |A^{(k)}|, |(A+u)^{(k)}| \} \geq |A|^b, \] for any finite $A \subset \mathbb Q$ and any non-zero $u \in \mathbb Q$. Here, $|A^{(k)}|$ denotes the $k$-fold product set $\{a_1\cdots a_k : a_1, \dots, a_k \in A \}$. Furthermore, our method of proof also gives the following $l_{\infty}$ sum-product estimate. For all $\gamma >0$ there exists a constant $C=C(\gamma...

We obtain a bounded generation theorem over $\mathcal O/\mathfrak a$, where $\mathcal O$ is the ring of integers of a number field and $\mathfrak a$ a general ideal of $\mathcal O$. This addresses a conjecture of Salehi-Golsefidy. Along the way, we obtain nontrivial bounds for additive character sums over $\mathcal O/\mathcal P^n$ for a prime ideal $\mathcal P$ with the aid of certain sum-product estimates.

In this paper we obtain a new sum-product estimate in prime fields. In particular, we show that if $A\subseteq \mathbb{F}_p$ satisfies $|A|\le p^{64/117}$ then $$ \max\{|A\pm A|, |AA|\} \gtrsim |A|^{39/32}. $$ Our argument builds on and improves some recent results of Shakan and Shkredov which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy $E^+(P)$ of some subset $P\subseteq A+A$. Our main novelty comes from reducing the estim...

We study a Szemer\'edi-Trotter type theorem in finite fields. We then use this theorem to obtain an improved sum-product estimate in finite fields.

It is well known that if S is a subset of the integers mod p, and if the second-largest Fourier coefficient is ``small'' relative to the largest coefficient, then the sumset S+S is much larger than S. We show in the present paper that if instead of having such a large ``spectral gap'' between the largest and second-largest Fourier coefficients, we had it between the kth largest and the (k+1)st largest, the same thing holds true, namely that |S+S| is appreciably larger than |S...

We prove some general estimates for exponential sums over subsets of finite fields which are definable in the language of rings. This generalizes both the classical exponential sum estimates over varieties over finite fields due to Weil, Deligne and others, and the result of Chatzidakis, van den Dries and Macintyre concerning the number of points of those definable sets. As a first application, there is no formula in the language of rings that defines for infinitely many prim...

These notes basically contain a material of two mini--courses which were read in G\"{o}teborg in April 2015 during the author visit of Chalmers & G\"{o}teborg universities and in Beijing in November 2015 during "Chinese--Russian Workshop on Exponential Sums and Sumsets". The article is a short introduction to a new area of Additive Combinatorics which is connected which so--called the higher sumsets as well as with the higher energies. We hope the notes will be helpful for a ...