Moscow Journal of Combinatorics and Number Theory

Convex sequences may have thin additive bases

Imre Z. Ruzsa and Dmitrii Zhelezov

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For a fixed c > 0 we construct an arbitrarily large set B of size n such that its sum set B + B contains a convex sequence of size c n 2 , answering a question of Hegarty.

Article information

Mosc. J. Comb. Number Theory, Volume 8, Number 1 (2019), 43-46.

Received: 1 December 2017
First available in Project Euclid: 3 December 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11B13: Additive bases, including sumsets [See also 05B10]

convex sequences sumset additive basis


Ruzsa, Imre Z.; Zhelezov, Dmitrii. Convex sequences may have thin additive bases. Mosc. J. Comb. Number Theory 8 (2019), no. 1, 43--46. doi:10.2140/moscow.2019.8.43.

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  • P. Hegarty, “Convex subsets of sumsets”, MathOverflow post, 2012, The original formulation is contrapositive to ours, which is more convenient to state.
  • N. Hegyvári, “On consecutive sums in sequences”, Acta Math. Hungar. 48:1-2 (1986), 193–200.
  • T. Schoen and I. D. Shkredov, “On sumsets of convex sets”, Combin. Probab. Comput. 20:5 (2011), 793–798.
  • I. D. Shkredov, “On sums of Szemerédi–Trotter sets”, Tr. Mat. Inst. Steklova 289:1 (2015), 318–327. In Russian; translated in Proc. Steklov Inst. Math. \bf289:1 (2015), 300–309.