On a conjecture of Fox–Kleitman and additive combinatorics
Résumé
Let D_k denote the maximum degree of regularity of the equation x_1 + • • • + x_k − y_1 − • • • − 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 recently been confirmed by T. Schoen and K. Taczala. Their proof is achieved by generalizing a theorem of Eberhard, Green and Manners 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} = lcm{i : i = 1,..., k − 1} is at least k − 1. This shows in a simple and explicit way that D_k behaves linearly in k.
Domaines
Mathématiques [math]Origine | Fichiers éditeurs autorisés sur une archive ouverte |
---|