On a conjecture of Fox–Kleitman and additive combinatorics

Abstract

Let $$D_k$$ denote the maximum degree of regularity of the equation $$x_1+\cdots +x_k-y_1-\cdots -y_k=b_k$$ as $$b_k$$ runs over the positive integers. The Fox and Kleitman conjecture, stating that $$D_k$$ should equal $$2k-1$$ , has been confirmed by Schoen and Taczala (Moscow J. Combin. Number Theory 7 (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 $$b_k = c_{k-1} = {{\,\mathrm{lcm}\,}}\{i :i = 1, \dots , k-1\}$$ is at least $$k-1$$ . This shows in a simple and explicit way that $$D_k$$ behaves linearly in k.