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January 17, 2026Forum of Mathematics Sigma0 citationsOpen Access

Asymmetric infinite sumsets in large sets of integers

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IKIoannis Kousek

Key Points

  • The study investigates the existence of infinite asymmetric sumsets within large integer sets with specific density conditions.
  • Defined set A with positive upper density.
  • Constructed infinite set B using pairwise combinations of integer elements.
  • Analyzed configurations under different density criteria and shifts.
  • Confirmed a conjecture regarding infinite sets in integers.
  • Demonstrated that sets with lower density > 1/2 include infinite configurations up to a shift.
  • Established that the threshold of 1/2 is optimal for density considerations.

Abstract

Abstract We show that for any set A N with positive upper density and any, m N, there exist an infinite set B N and some t N so that \mb₁ + b₂ b₁, b₂ B\ {and\ b₁1/2 contains such configurations up to a shift. We show that the value 1/2 is optimal and obtain analogous results for values of upper density and when no shift is allowed.

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Cite This Study

Ioannis Kousek (2026) studied this question.

synapsesocial.com/papers/696b25a9d2a12237a9348fd8https://doi.org/10.1017/fms.2025.10157
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