A duality principle for the neighbourhood of manifolds and applications to counting rational points near curves

Abstract

In this paper, we establish a duality principle relating the count of rationals near a manifold to a sparse counting problem for the dual. As applications, we give a short, self-contained proof of sharp bounds for planar curves under minimal regularity conditions and similarly, sharp upper bounds for non- degenerate space curves, again under minimal curvature conditions. This work recovers and extends the works of Beresnevich, Dickinson and Velani and of Vaughan and Velani on planar curves and independently proves and extends the results of Chen, Seeger, Srivastava and Technau on space curves, while providing a structural picture well-suited for generalisations. The re- sults and methods of this work precede the emergence of highly capable AI models.