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.
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Keywords: Sidon sets, pseudorandomness, equidistribution, partition regularity
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@article{JTNB_2023__35_1_115_0,
author = {Miquel Ortega and Sean Prendiville},
title = {Extremal {Sidon} {Sets} are {Fourier} {Uniform,} with {Applications} to {Partition} {Regularity}},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {115--134},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {35},
number = {1},
year = {2023},
doi = {10.5802/jtnb.1239},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1239/}
}
TY - JOUR AU - Miquel Ortega AU - Sean Prendiville TI - Extremal Sidon Sets are Fourier Uniform, with Applications to Partition Regularity JO - Journal de théorie des nombres de Bordeaux PY - 2023 SP - 115 EP - 134 VL - 35 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1239/ DO - 10.5802/jtnb.1239 LA - en ID - JTNB_2023__35_1_115_0 ER -
%0 Journal Article %A Miquel Ortega %A Sean Prendiville %T Extremal Sidon Sets are Fourier Uniform, with Applications to Partition Regularity %J Journal de théorie des nombres de Bordeaux %D 2023 %P 115-134 %V 35 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1239/ %R 10.5802/jtnb.1239 %G en %F JTNB_2023__35_1_115_0
Miquel Ortega; Sean Prendiville. Extremal Sidon Sets are Fourier Uniform, with Applications to Partition Regularity. Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 115-134. doi : 10.5802/jtnb.1239. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1239/
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