On a conjecture of Fox–Kleitman and additive combinatorics

Abstract

Let Dk denote the maximum degree of regularity of the equation x1+ ⋯ + xk- y1- ⋯ - yk= bk as bk runs over the positive integers. The Fox and Kleitman conjecture, stating that Dk should equal 2 k- 1 , has been confirmed by Schoen and Taczala (Moscow J. Combin. Number Theory7 (2017) 79–93). Their proof is achieved by generalizing a theorem of Eberhard et al. (Ann. Math. 180 (2014) 621–652) on sets with doubling constant less than 4. Using much simpler methods and a result of Lev in additive combinatorics, our main result here is that the degree of regularity of the same equation for the specific value bk=ck-1=lcm{i:i=1,⋯,k-1} is at least k- 1. This shows in a simple and explicit way that Dk behaves linearly in k.

Publication Title

Proceedings of the Indian Academy of Sciences: Mathematical Sciences

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