Difference Sets of Null Density Subsets of N

Abstract

Let , and for any , . If is positive, then B is considered as a large set with . Its difference set has both high density and rich structure. The set with is also relatively large and it is a long standing conjecture that like sets with positive upper density they have arithmetic progression of arbitrary length. Here we show their difference set may not be substantial; for any there exists such that and .

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D. Dastjerdi and M. Hosseini, "Difference Sets of Null Density Subsets of N," Advances in Pure Mathematics, Vol. 2 No. 3, 2012, pp. 195-199. doi: 10.4236/apm.2012.23027.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] I. Z. Rusza, “On Difference-Sequences,” Acta Arithmetica, Vol. 25, 1974, pp. 151-157. [2] V. Bergelson and N. Hindman, “Additive and Multiplica- tive Ramsey Theorems in Some Elementary Results,” Combinatorics, Probability and Computing, Vol. 2, 1993, pp. 221-241. doi:10.1017/S0963548300000638 [3] V. Bergelson, “Partition Regular Structures Contained in Large Sets Are Abundant,” Journal of Combinatorial Theory, Series A, Vol. 93, No. 1, 2001, pp. 18-36.doi:10.1006/jcta.2000.3061 [4] T. C. Brown and A. R. Freedman, “Arithmetic Progressions in Lacunary Sets,” Rocky Mountain Journal of Mathematics, Vol. 17, No. 3, 1987, pp. 587-596. doi:10.1216/RMJ-1987-17-3-587 [5] V. Bergelson, N. Hindman and R. McCutchen, “Notions of Size and Combinatorial Properties of Quotient Sets in Semigroups,” Topology Proceedings, Vol. 23, 1998, pp. 23-60. [6] E. Szemerdi, “On Sets of Integers Containing No k Ele- ments in Arithmetic Progression,” Acta Arithmetica, Vol. 27, 1975, pp. 199-245. [7] P. Erd?s, “Problems and Results in Combinatorial Number Theory,” Astrisque, 1975, pp. 295-310. [8] B. Green and T. Tao, “The Primes Contain Arbitrarily Long Arithmetic Progressions,” Annals of Mathematics, Vol. 167, No. 2, 2004, pp. 481-547. doi:10.4007/annals.2008.167.481