A Purely Combinatorial Approach to Simultaneous Polynomial Recurrence modulo 1 Ernie Croot Neil Lyall

A Purely Combinatorial Approach to Simultaneous Polynomial Recurrence modulo 1

Using purely combinatorial means we obtain results on simultaneous Diophantine approximation modulo 1 for systems of polynomials with real coefficients and no constant term.

A Quantitative Result on Diophantine Approximation for Intersective Polynomials

In this short note, we closely follow the approach of Green and Tao to extend the best known bound for recurrence modulo 1 from squares to the largest possible class of polynomials. The paper concludes with a brief discussion of a consequence of this result for polynomial structures in sumsets and limitations of the method.

Combinatorial Proof of the Hot Spot Theorem

SUMS OF THE FORM 1 / x k 1 + · · · + 1 / x kn MODULO A PRIME

Using a sum-product result due to Bourgain, Katz, and Tao, we show that for every 0 < 2 ≤ 1, and every integer k ≥ 1, there exists an integer N = N(2, k), such that for every prime p and every residue class a (mod p), there exist positive integers x1, ..., xN ≤ p satisfying a ≡ 1 x1 + · · ·+ 1 xN (mod p).

Reciprocal Power Sums Modulo a Prime

Using a sum-product result due to Bourgain, Katz, and Tao, we show that for every 0 < ǫ ≤ 1, and every power k ≥ 1, there exists an integer N = N(ǫ, k), such that for every prime p and every residue class a (mod p), there exist positive integers x1, ..., xN ≤ p ǫ satisfying a ≡ 1 x1 + · · · + 1 xN (mod p). This extends a result of I. Shparlinski [5].