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Institut Henri Poincaré


À la redécouverte des points rationnels

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Séminaire

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Badly distributed sets lying on algebraic varieties over global fields

Paredes

Let \(N\) be a natural number and let \(S\) be a subset of \(N^{n}\). A result of Walsh states that if \(S\) is badly distributed at the level of residue classes modulo \(p\) for various primes \(p\), then \(S\) is “small”, or it possesses some “algebraic structure”, namely a positive proportion of \(S\) lies in an hypersurface defined over the rational numbers of small degree. Moreover, Walsh also proved that a positive proportion of any such \(S\) always lies in an hypersurface defined over the rational numbers of small degree, but now the degree depending on \(N\).

In this talk, we will show that these results can be generalized to sets \(S\) lying in projective varieties defined over global fields, and that they can be used to obtain some diophantine applications. Part of this talk is work in progress with J. Menconi and R. Sasyk.