Sparse distribution of lattice points in annular regions

Abstract

This paper is inspired by Richards' work on large gaps between sums of two squares [10]. It is shown in [10] that there exist arbitrarily large values of λ and μ, where μ≥Clog⁡λ, such that intervals [λ,λ+μ] do not contain any sums of two squares. Geometrically, these gaps between sums of two squares correspond to annuli in R2 that do not contain any integer lattice points. A major objective of this paper is to investigate the sparse distribution of integer lattice points within annular regions in R2. Specifically, we establish the existence of annuli {x∈R2:λ≤|x|2≤λ+κ} with arbitrarily large λ and κ≥Cλs for 0<s<14, satisfying that any two integer lattice points within any one of these annuli must be sufficiently far apart. This result is sharp, as such a property ceases to hold at and beyond the threshold s=14. Furthermore, we extend our analysis to include the sparse distribution of lattice points in spherical shells in R3.