# The structure of popular difference sets

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## Abstract

We show that the set of popular differences of a large subset of ℤ_{ N } does not always contain the complete difference set of another large set. For this purpose we construct a so-called niveau set, which was first introduced by Ruzsa in [Ruz87] and later used in [Ruz91] to show that there exists a large subset of ℤ_{ N } whose sumset does not contain any long arithmetic progressions. In this paper we make substantial use of measure concentration results on the multi-dimensional torus and Esseen’s Inequality.

## Keywords

Independent Random Variable Arithmetic Progression Total Variation Distance Joint Distribution Function Standard Normal Distribution Function
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