Thái Hoàng Lê

Problems and Results on Intersective Sets

By intersective set we mean a set H ⊂ Z having the property that it intersects the difference set A − A of any dense subset A of the integers. By analogy between the integers and the ring of polynomials over a finite field, the notion of intersective sets also makes sense in the latter setting. We give a survey of methods used to study intersective sets, known results and open problems in both settings.

On primitive elements in finite fields of low characteristic

We discuss the problem of constructing a small subset of a finite field containing primitive elements of the field. Given a finite field,

Polynomial Configurations in the Primes

\mathbb{F}_{q^n}

Uniform dilations in higher dimensions

, small

A note on character sums in finite fields

q

Additive bases in groups

and large

Deterministic Extractors for Additive Sources: Extended Abstract

n

Intersective polynomials and the primes

, 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 the integer case and might be of independent interest.

Green–Tao theorem in function fields

The Bergelson-Leibman theorem states that if P_1, ..., P_k are polynomials with integer coefficients, then any subset of the integers of positive upper density contains a polynomial configuration x+P_1(m), ..., x+P_k(m), where x,m are integers. Various generalizations of this theorem are known. Wooley and Ziegler showed that the variable m can in fact be taken to be a prime minus 1, and Tao and Ziegler showed that the Bergelson-Leibman theorem holds for subsets of the primes of positive relative upper density. Here we prove a hybrid of the latter two results, namely that the step m in the Tao-Ziegler theorem can be restricted to the set of primes minus 1.

SUMS OF PRODUCTS OF FRACTIONAL PARTS

A theorem of Glasner says that if