## 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.

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Wolf, J. The structure of popular difference sets.
*Isr. J. Math.* **179**, 253–278 (2010). https://doi.org/10.1007/s11856-010-0081-2

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DOI: https://doi.org/10.1007/s11856-010-0081-2

### Keywords

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