Counting Rational Points In Non-Isotropic Neighborhoods of Manifolds

2024/07/03 by Srivastava, Rajula · 3 citations
#11D75 #11J13 #11J25 #11J83 #11K55 #11K60 #42B20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2407.03078

Abstract

In this manuscript, we initiate the study of the number of rational points with bounded denominators, contained in a non-isotropic δ₁×…× δ_R neighborhood of a compact submanifold M of codimension R in ℝ^M. We establish an upper bound for this counting function which holds when M satisfies a strong curvature condition, first introduced by Schindler-Yamagishi in \citeschindler2022density. Further, even in the isotropic case when δ₁=…=δ_R=δ, we obtain an asymptotic formula which holds beyond the range of distance to M established in \citeschindler2022density. Our result is also a generalization of the work of J.J. Huang \citehuangduke for hypersurfaces. As an application, we establish for the first time an upper bound for the Hausdorff dimension of the set of weighted simultaneously well approximable points on a manifold M satisfying the strong curvature condition, which agrees with the lower bound obtained by Allen-Wang in \citeallen2022note. Moreover, for R>1, we obtain a new upper bound for the number of rational points on M, which goes beyond the bound in an analogue of Serre's dimension growth conjecture for submanifolds of ℝ^M .

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