Norbert Hegyvári

Explicit constructions of extractors and expanders

We investigate 2-variable expanders and 3-source extractors in prime fields. We extend previous results of J. Bourgain.

On Hilbert Cubes in Certain Sets

A set of the form {d + Σi εi a : εi = 0 or 1, Σi εi < ∞} (where d is a non-negative integer, a1, a2, ... are positive integers) is called a Hilbert cube. If {a1, a2, ...} is a finite set of, say, k elements, then it called a k-cube, while if {a1, a2, ...} is infinite, then the cube is said to be an infinite cube. As a partial answer to a question of Brown, Erdös and Freedman, an upper bound is given for the size of a Hilbert cube contained in the set of the squares not exceeding n. Estimates of Gaussian sums, Gallaghers “large sieves” and a result of Olson play a crucial rule in the proof. Hilbert cubes in other special sets are also studied.

On the Completeness of an Exponential Type Sequence

We investigate the Birchs sequence YK = pαqβ ¦ p, q > 1, α, β N0, 0 ≦ β ≦ K giving a partial answer for a question of P. Erdös.

A structure result for bricks in Heisenberg groups

Abstract We show that for a sufficiently big brick B of the ( 2 n + 1 ) -dimensional Heisenberg group H n over the finite field F p , the product set B ⋅ B contains at least | B | / p many cosets of some non-trivial subgroup of H n .

On iterated difference sets in groups

We extend a result of Stewart, Tijdeman and Ruzsa on iterated difierence sequences to groups. We give a complete answer for abelian groups, and apart from a constant, we give the best estimate for non-abelian ones.

Conditional expanding bounds for two-variable functions over prime fields

In this paper we provide in F p expanding lower bounds for two variables functions f ( x , y ) in connection with the product set or the sumset. The sum-product problem has been immensely studied in the recent past. A typical result in F p ? is the existence of Δ ( α ) 0 such that if | A | ? p α then max ( | A + A | , | A ? A | ) ? | A | 1 + Δ ( α ) , Our aim is to obtain analogous results for related pairs of two-variable functions f ( x , y ) and g ( x , y ) : if | A | ? | B | ? p α then max ( | f ( A , B ) | , | g ( A , B ) | ) ? | A | 1 + Δ ( α ) for some Δ ( α ) 0 .

Complete sequences in N 2

Folkmans theorem states that ifA={a1 n1/2+ϵ, whereA(n)=∑ai≤n1), thenP(A)={ai1+ai2+ · · ·+ais: i1<i2< · · ·<is;s∈N;aij∈A}contains an infinite arithmetic progression. We shall consider the analogue of Folkmans theorem inN2, and some related questions are also investigated.

Distribution of residues in approximate subgroups of _

We extend a result due to Bourgain on the uniform distribution of residues by proving that subsets of the type

Expansion for cubes in the Heisenberg group

f(I)\cdot H

On additive and multiplicative Hilbert cubes

is equidistributed (as