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arXiv · 0711.3201

Polynomial largeness of sumsets and totally ergodic sets

A. Fish
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Abstract

We prove that a sumset of a TE subset of (\N) (these sets can be viewed as "aperiodic" sets) with a set of positive upper density intersects a set of values of any polynomial with integer coefficients., i.e. for any (A \subset \N ) a TE set, for any (p(n) \in \Z[n]: °{p(n)} > 0, p(n) \to_{n \to \infty} \infty ) and any subset (B \subset \N ) of positive upper density we have (R_p = A+B \cap \{p(n) | n \in \N \} \neq \emptyset). For (A ) a WM set (subclass of TE sets) we prove that (R_p ) has lower density 1. In addition we obtain a generalization of the latter result to the case of several polynomials and several WM sets.

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A. Fish. 2007-11-20. Polynomial largeness of sumsets and totally ergodic sets. https://arxiv.org/abs/0711.3201

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