The VC-dimension of quadratic residues in finite fields

Let $\mathbb{F}_q$ be a finite field of order $q$. Iosevich and Rudnev (2005) proved that for any set $A\subset \mathbb{F}_q^d$, if $|A|\gg q^{\frac{d+1}{2}}$, then the distance set $\Delta(A)$ contains a positive proportion of all distances. Although this result is sharp in odd dimensions, it is conjectured that the right exponent should be $\frac{d}{2}$ in even dimensions. During the last 15 years, only some improvements have been made in two dimensions, and the conjecture ...

We consider the linear vector space formed by the elements of the finite fields $\mathbb{F}_q$ with $q=p^r$ over $\mathbb{F}_p$. Let ${a_1,\ldots,a_r}$ be a basis of this space. Then the elements $x$ of $\mathbb{F}_q$ have a unique representation in the form $\sum_{j=1}^r c_ja_j$ with $c_j\in\mathbb{F}_p$. Let $D$ be a subset of $\mathbb{F}_p$. We consider the set $W_D$ of elements of $\mathbb{F}_q$ such that $c_j\in D$ for all $j=1,\ldots,r$. We give an estimate for the numb...

A long standing problem has been to develop "good" binary linear codes to be used for error-correction. This paper investigates in some detail an attack on this problem using a connection between quadratic residue codes and hyperelliptic curves. One question which coding theory is used to attack is: Does there exist a c<2 such that, for all sufficiently large $p$ and all subsets S of GF(p), we have |X_S(GF(p))| < cp?

We prove that if $A \subset {\Bbb F}_q$ is such that $$|A|>q^{{1/2}+\frac{1}{2d}},$$ then $${\Bbb F}_q^{*} \subset dA^2=A^2+...+A^2 d \text{times},$$ where $$A^2=\{a \cdot a': a,a' \in A\},$$ and where ${\Bbb F}_q^{*}$ denotes the multiplicative group of the finite field ${\Bbb F}_q$. In particular, we cover ${\Bbb F}_q^{*}$ by $A^2+A^2$ if $|A|>q^{{3/4}}$. Furthermore, we prove that if $$|A| \ge C_{size}^{\frac{1}{d}}q^{{1/2}+\frac{1}{2(2d-1)}},$$ then $$|dA^2| \ge q \cdot \...

Let $A$ be a simple abelian variety of dimension $g$ over the field $\mathbb{F}_q$. The paper provides improvements on the Weil estimates for the size of $A(\mathbb{F}_q)$. For an arbitrary value of $q$ we prove $(\lfloor(\sqrt{q}-1)^2 \rfloor + 1)^g \leqslant A(\mathbb{F}_q) \leqslant (\lceil(\sqrt{q}+1)^2 \rceil - 1)^{g}$ holds with finitely many exceptions. We compute improved bounds for various small values of $q$. For instance, the Weil bounds for $q=3,4$ give a trivial ...

We discuss the problem of constructing a small subset of a finite field containing primitive elements of the field. Given a finite field, $\mathbb{F}_{q^n}$, small $q$ and large $n$, we show that the set of all low degree polynomials contains the expected number of primitive elements. The main theorem we prove is a bound for character sums over short intervals in function fields. Our result is unconditional and slightly better than what is known (conditionally under GRH) in...

Suppose that an infinite set $A$ occupies at most $\frac{1}{2}(p+1)$ residue classes modulo $p$, for every sufficiently large prime $p$. The squares, or more generally the integer values of any quadratic, are an example of such a set. By the large sieve inequality the number of elements of $A$ that are at most $X$ is $O(X^{1/2})$, and the quadratic examples show that this is sharp. The simplest form of the inverse large sieve problem asks whether they are the only examples. W...

In this paper we prove some results on sum-product estimates over arbitrary finite fields. More precisely, we show that for sufficiently small sets $A\subset \mathbb{F}_q$ we have \[|(A-A)^2+(A-A)^2|\gg |A|^{1+\frac{1}{21}}.\] This can be viewed as the Erd\H{o}s distinct distances problem for Cartesian product sets over arbitrary finite fields. We also prove that \[\max\{|A+A|, |A^2+A^2|\}\gg |A|^{1+\frac{1}{42}}, ~|A+A^2|\gg |A|^{1+\frac{1}{84}}.\]

Let $\mathbb{F}_q$ be a finite field of order $q$. In this paper, we study the distribution of rectangles in a given set in $\mathbb{F}_q^2$. More precisely, for any $0<\delta\le 1$, we prove that there exists an integer $q_0=q_0(\delta)$ with the following property: if $q\ge q_0$ and $A$ is a multiplicative subgroup of $\mathbb{F}^*_q$ with $|A|\ge q^{2/3}$, then any set $S\subset \mathbb{F}_q^2$ with $|S|\ge \delta q^2$ contains at least $\gg \frac{|S|^4|A|^2}{q^5}$ rectang...

We obtain quantitative versions of the Balog-Szemeredi-Gowers and Freiman theorems in the model case of a finite field geometry F_2^n, improving the previously known bounds in such theorems. For instance, if A is a subset of F_2^n such that |A+A| <= K|A| (thus A has small additive doubling), we show that there exists an affine subspace V of F_2^n of cardinality |V| >> K^{-O(\sqrt{K})} |A| such that |A \cap V| >> |V|/2K. Under the assumption that A contains at least |A|^3/K qu...