Counting lattice points and o-minimal structures

2012/10/22 by Barroero, Fabrizio, Widmer, Martin · 2 citations
#03C64 (Primary) 11P21 #03C98 #11H06 #28A75 #52C07 (Secondary) #FOS: Mathematics #Logic (math.LO) #Metric Geometry (math.MG) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1210.5943

Abstract

Let Λ be a lattice in \Rⁿ, and let Z⊆ \R^m+n be a definable family in an o-minimal structure over \R. We give sharp estimates for the number of lattice points in the fibers Z_T=x∈ \Rⁿ: (T,x)∈ Z. Along the way we show that for any subspace Σ⊆\Rⁿ of dimension j>0 the j-volume of the orthogonal projection of Z_T to Σ is, up to a constant depending only on the family Z, bounded by the maximal j-dimensional volume of the orthogonal projections to the j-dimensional coordinate subspaces.

Cited by

• Lower bounds on heights of odd degree points of hyperelliptic curves
• Integral points of bounded height on quintic del Pezzo surfaces over number fields

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