# The structure of popular difference sets

Article

First Online:

- 111 Downloads
- 3 Citations

## 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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- [AK94]W. Albers and W. C. M. Kallenberg,
*A simple approximation to the bivariate normal distribution with large correlation coefficient*, Journal of Multivariate Analysis**49**(1994), 87–96.zbMATHCrossRefMathSciNetGoogle Scholar - [AS00]N. Alon and J. Spencer,
*The Probabilistic Method*, Wiley-Interscience [John Wiley & Sons], New York, 2000.zbMATHCrossRefGoogle Scholar - [Ber41]A. C. Berry,
*The accuracy of the Gaussian approximation to the sum of independent variates*, Transactions of the American Mathematical Society**49**(1941), 122–136.zbMATHMathSciNetGoogle Scholar - [Ber45]H. Bergstrom,
*On the central limit theorem in the case*ℝ^{k},*k*> 1, Skand. Aktuarietidskr.**2**(1945), 106–127.MathSciNetGoogle Scholar - [DT97]M. Drmota and R. F. Tichy,
*Sequences, Discrepancies and Applications*, Springer-Verlag, Berlin, 1997.zbMATHGoogle Scholar - [Ess45]C-G. Esseen,
*Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law*, Acta Mathematica**77**(1945), 1–125.zbMATHCrossRefMathSciNetGoogle Scholar - [Gre03]B. J. Green,
*Some constructions in the inverse spectral theory of cyclic groups*, Combinatorics, Probability and Computing**12**(2003), 127–138.zbMATHCrossRefMathSciNetGoogle Scholar - [Gre05]B. J. Green,
*Finite field models in additive combinatorics*, in*Surveys in combinatorics 2005*, London Math. Soc. Lecture Note Ser., vol. 327, Cambridge University Press, Cambridge, 2005, pp. 1–27.CrossRefGoogle Scholar - [GR05]B. J. Green and I. Ruzsa,
*Sum-free sets in abelian groups*, Israel Journal of Mathematics**147**(2005), 157–189.zbMATHCrossRefMathSciNetGoogle Scholar - [Har66]L. H. Harper,
*Optimal numberings and isoperimetric problems on graphs*, Journal of Combinatorial Theory**1**(1966), 385–393.zbMATHCrossRefMathSciNetGoogle Scholar - [JŁR00]S. Janson, T. Łuczak and A. Rucinski,
*Random Graphs*, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience [John Wiley & Sons], New York, 2000.zbMATHGoogle Scholar - [Kle66]D. Kleitman,
*On a combinatorial conjecture by Erdös*, Journal of Combinatorial Theory**1**(1966), 209–214.zbMATHCrossRefMathSciNetGoogle Scholar - [Led01]M. Ledoux,
*The concentration of measure phenomenon*, AMS Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, Providence, RI, 2001.zbMATHGoogle Scholar - [LŁS01]V. Lev, T. Łuczak and T. Schoen,
*Sum-free sets in abelian groups*, Israel Journal of Mathematics**125**(2001), 347–367.zbMATHCrossRefMathSciNetGoogle Scholar - [McD89]C. McDiarmid,
*On the method of bounded differences*, in*Surveys in combinatorics, 1989 (Norwich, 1989)*, London Math. Soc. Lecture Note Ser., vol. 141, Cambridge University Press, Cambridge, 1989, pp. 148–188.Google Scholar - [NP73]H. Niederreiter and W. Philipp,
*Berry-Esseen bounds and a theorem of Erdös and Turán on uniform distribution mod 1*, Duke Mathematical Journal**40**(1973), 633–649.zbMATHCrossRefMathSciNetGoogle Scholar - [Ruz87]I. Z. Ruzsa,
*Essential components*, Proceedings of the London Mathematical Society. Third Series**54**(1987), 38–56.zbMATHCrossRefMathSciNetGoogle Scholar - [Ruz91]I. Z. Ruzsa,
*Arithmetic progressions in sumsets*, Acta Arithmetica**2**(1991), 191–202.MathSciNetGoogle Scholar - [Sad66]S. M. Sadikova,
*Two-dimensional analogues of an inequality of Esseen with applications to the Central Limit Theorem*, Theory of Probability and Its Applications**11**(1966), 325–335.CrossRefGoogle Scholar - [San08]T. Sanders,
*Popular difference sets*, Available at http://arxiv.org/abs/0807.5106, 2008. - [Shi84]A. N. Shiryayev,
*Probability, Graduate Texts in Mathematics*, vol. 95, Springer-Verlag, New York, 1984.Google Scholar

## Copyright information

© Hebrew University Magnes Press 2010