On asymptotic formulae in some sum-product questions

Let $p$ be a prime and $n$ a positive integer such that $\sqrt{\frac p2} + 1 \leq n \leq \sqrt{p}$. For any arithmetic progression $A$ of length $n$ in $\mathbb{F}_p$, we establish an asymptotic formula for the number of directions determined by $A \times A \subset \mathbb{F}_p^2$. The key idea is to reduce the problem to counting the number of solutions to the bilinear Diophantine equation $ad+bc=p$ in variables $1\le a,b,c,d\le n$; our asymptotic formula for the number of s...

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 ...

We show that under the assumption of a 24-term version of Fermat's Last Theorem, there exists an absolute constant c > 0 such that if S is a set of n > n_0 positive integers satisfying |S.S| < n^(1+c), then the sumset S.S satisfies |S+S| >> n^2. In other words, we prove a weak form of the Erdos-Szemeredi sum-product conjecture, conditional on an extension of Fermat's Last Theorem. Unconditionally, we prove this theorem for when S is a set of n monic polynomials. We also prove...

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 develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables which complements works by T. Cochrane and Z. Zheng on the single variable case. As an application, for $n\geq 2$, a fixed natural number, we obtain an asymptotic formula for the (weighted) number of solutions of quadratic congruences of the form $x_1^2+x_2^2+...+x_n^2\equiv x_{n+1}^2\bmod{p^m}$ in small boxes, thus establishing an equidistributio...

The main results of this paper concern growth in sums of a $k$-convex function $f$. Firstly, we streamline the proof of a growth result for $f(A)$ where $A$ has small additive doubling, and improve the bound by removing logarithmic factors. The result yields an optimal bound for \[ |2^k f(A) - (2^k-1)f(A)|. \] We also generalise a recent result of Hanson, Roche-Newton and Senger, by proving that for any finite $A\subset \mathbb{R}$ \[ | 2^k f(sA-sA) - (2^k-1) f(sA-sA)| \g...

Let $\mathbb{F}_q$ be a finite field of order $q$, where $q$ is a power of a prime. For a set $A \subset \mathbb{F}_q$, under certain structural restrictions, we prove a new explicit lower bound on the size of the product set $A(A + 1)$. Our result improves on the previous best known bound due to Zhelezov and holds under more relaxed restrictions.

We provide estimates for sums of the form \[\left|\sum_{a\in A}\sum_{b\in B}\sum_{c\in C}\chi(a+b+c)\right|\] and \[\left|\sum_{a\in A}\sum_{b\in B}\sum_{c\in C}\sum_{d\in D}\chi(a+b+cd)\right|\] when $A,B,C,D\subset \mathbb F_p$, the field with $p$ elements and $\chi$ is a non-trivial multiplicative character modulo $p$.

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 ...

In additive combinatorics, Erd\"{o}s-Szemer\'{e}di Conjecture is an important conjecture. It can be applied to many fields, such as number theory, harmonic analysis, incidence geometry, and so on. Additionally, its statement is quite easy to understand, while it is still an open problem. In this dissertation, we investigate the Erd\"{o}s-Szemer\'{e}di Conjecture and its relationship with several well-known results in incidence geometry, such as the Szemer\'{e}di-Trotter Incid...