Abstract
Sharpening (a particular case of) a result of Szemerédi and Vu [4] and extending earlier results of Sárközy [3] and ourselves [2], we find, subject to some technical restrictions, a sharp threshold for the number of integer sets needed for their sumset to contain a block of consecutive integers, whose length is comparable with the lengths of the set summands.
A corollary of our main result is as follows. Let k,l≥1 and n≥3 be integers, and suppose that A 1,…,A k ⊆[0,l] are integer sets of size at least n, none of which is contained in an arithmetic progression with difference greater than 1. If k≥2⌈(l−1)/(n−2)⌉, then the sumset A 1+⋅⋅⋅+A k contains a block of at least k(n−1)+1 consecutive integers.
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References
V. Lev, Addendum to “Structure theorem for multiple addition”, J. Number Theory, 65 (1997), 96–100.
V. Lev, Optimal representations by sumsets and subset sums, J. Number Theory, 62 (1997), 127–143.
A. Sárközy, Finite addition theorems, I, J. Number Theory, 32 (1989), 114–130.
E. Szemerédi and V. Vu, Long arithmetic progressions in sumsets: thresholds and bounds, J. Amer. Math. Soc., 19 (2006), 119–169.
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The author gratefully acknowledges the support of the Georgia Institute of Technology and the Fields Institute, which he was visiting while conducting his research.
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Lev, V.F. Consecutive integers in high-multiplicity sumsets. Acta Math Hung 129, 245–253 (2010). https://doi.org/10.1007/s10474-010-0026-6
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DOI: https://doi.org/10.1007/s10474-010-0026-6