Moment formulas of Siegel transforms with congruence conditions in dimension 2

TL;DR

This paper derives moment formulas for Siegel transforms under congruence conditions in two dimensions, leading to new results in lattice point counting and Diophantine approximation with congruence constraints.

Contribution

It provides the first and second moment formulas for Siegel transforms with congruence conditions in dimension two, extending classical lattice point counting results.

Findings

Derived analog of Schmidt's random counting theorem.

Established a quantitative Khintchine theorem with congruence constraints.

Enhanced understanding of lattice point distributions under modular restrictions.

Abstract

We compute the first and second moment formulas for Siegel transforms related to problems counting primitive lattice points in the real plane with congruence conditions. As applications, we derive an analog of Schmidt's random counting theorem and the quantitative Khintchine theorem for irrational numbers, approximated by rational numbers $p/q$, where we place a congruence-conditional constraint on the vector $(p,q)$.