Extremal Sidon Sets are Fourier Uniform, with Applications to Partition Regularity

Ortega, Miquel; Prendiville, Sean

Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 1, pp. 115-134

Generalising results of Erdős–Freud and Lindström, we prove that the largest Sidon subset of a bounded interval of integers is equidistributed in Bohr neighbourhoods. We establish this by showing that extremal Sidon sets are Fourier-pseudorandom, in that they have no large non-trivial Fourier coefficients. As a further application we deduce that, for any partition regular equation in five or more variables, every finite colouring of an extremal Sidon set has a monochromatic solution.

En généralisant des résultats d’Erdős–Freud et Lindström, nous prouvons que le plus grand sous-ensemble de Sidon d’un intervalle d’entiers borné est équidistribué dans des voisinages de Bohr. Nous le faisons en montrant que les ensembles de Sidon extrémaux sont Fourier-pseudo-aléatoires, dans le sens qu’ils n’ont pas de coefficients de Fourier grands non triviaux. Comme application, nous en déduisons que pour une equation régulière à cinq variables et plus, toute coloration finie d’un ensemble extrémal de Sidon a une solution monochromatique.

Classification : 11B30, 11B25
Keywords: Sidon sets, pseudorandomness, equidistribution, partition regularity

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