The largest $(k,\ell )$-sum-free subsets
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- by Yifan Jing and Shukun Wu PDF
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Abstract:
Let $\mathscr {M}_{(2,1)}(N)$ be the infimum of the largest sum-free subset of any set of $N$ positive integers. An old conjecture in additive combinatorics asserts that there is a constant $c=c(2,1)$ and a function $\omega (N)\to \infty$ as $N\to \infty$, such that $cN+\omega (N)<\mathscr {M}_{(2,1)}(N)<(c+o(1))N$. The constant $c(2,1)$ is determined by Eberhard, Green, and Manners, while the existence of $\omega (N)$ is still wide open.
In this paper, we study the analogous conjecture on $(k,\ell )$-sum-free sets and restricted $(k,\ell )$-sum-free sets. We determine the constant $c(k,\ell )$ for every $(k,\ell )$-sum-free sets, and confirm the conjecture for infinitely many $(k,\ell )$.
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Additional Information
- Yifan Jing
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana Illinois
- MR Author ID: 1307767
- ORCID: 0000-0002-9954-7182
- Email: yifanjing17@gmail.com
- Shukun Wu
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana Illinois
- MR Author ID: 1404577
- Email: shukunw2@illinois.edu
- Received by editor(s): January 18, 2020
- Received by editor(s) in revised form: July 11, 2020, September 7, 2020, and January 9, 2021
- Published electronically: April 27, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 5163-5189
- MSC (2020): Primary 11B30; Secondary 11K70, 05D10
- DOI: https://doi.org/10.1090/tran/8385
- MathSciNet review: 4273189
Dedicated: This paper is dedicated to the memory of Jean Bourgain