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Journal Articles Proceedings Mathematical Sciences Year : 2019

On a conjecture of Fox–Kleitman and additive combinatorics

S D Adhikari
  • Function : Author
R Balasubramanian
  • Function : Author
D J Grynkiewicz
  • Function : Author

Abstract

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.
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hal-04122786 , version 1 (08-06-2023)

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S D Adhikari, R Balasubramanian, Shalom Eliahou, D J Grynkiewicz. On a conjecture of Fox–Kleitman and additive combinatorics. Proceedings Mathematical Sciences, 2019, 129 (4), pp.43. ⟨10.1007/s12044-019-0488-6⟩. ⟨hal-04122786⟩
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