On a Question of Sárközy

On a Question of Sárközy on Gaps of Product Sequences

Motivated by a question of Sárközy, we study the gaps in the product sequence B = A · A = {b1 < b2 < · · · } of all products aiaj with ai, aj ∈ A when A has upper Banach density α > 0. We prove that there are infinitely many gaps bn+1 − bn ¿ α−3 and that for t ≥ 2 there are infinitely many t-gaps bn+t − bn ¿ t2α−4. Furthermore we prove that these estimates are best possible. We also discuss a r...

On Some Problems of Gyarmati and Sárközy

In a recent paper, for “large” (but otherwise unspecified) subsets A,B, C,D of Fq, Gyarmati and Sárközy (2008) showed the solvability of the equations a + b = cd, and ab + 1 = cd with a ∈ A, b ∈ B, c ∈ C, d ∈ D. They asked whether one can extend these results to every k ∈ N in the following way: for large subsets A,B, C,D of Fq, there are a1, . . . , ak, a1, . . . , ak ∈ A, b1, . . . , bk, b1, ...

D-sets and a Sárközy Theorem for Countable Fields

We establish that if F is a countable field of characteristic p, U is a unitary action of F` on a Hilbert space H , and P is an essential idempotent ultrafilter on F then for every polynomial r : F → F` with r(0) = 0 one has P-limg Ur(g)f = Prf weakly, where Pr is the projection onto the closed subspace Kr = { f ∈ H : {Ur(g)f}g∈F is precompact in the strong topology } . We then derive combinato...

Investigating Embedded Question Reuse in Question Answering

The investigation presented in this paper is a novel method in question answering (QA) that enables a QA system to gain performance through reuse of information in the answer to one question to answer another related question. Our analysis shows that a pair of question in a general open domain QA can have embedding relation through their mentions of noun phrase expressions. We present methods f...

A Question and Answer Session on the Meaning of Being Human , with Dr. Gholamreza Aavani