Moment formulas of Siegel transforms with congruence conditions in dimension 2

2025/07/08 by Jiyoung Han, Seul Bee Lee, Han, Jiyoung +1
Mathematics · #22E40 #Advanced Combinatorial Mathematics #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Primary 11P21 #Secondary 11K60

paper · pdf · doi:10.48550/arxiv.2507.05905

openalex publication_date 2025/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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).

Related

• Mean value theorems for the S-arithmetic primitive Siegel transforms
• Rogers' mean value theorem for S-arithmetic Siegel transform and applications to the geometry of numbers
• Second moment of the light-cone Siegel transform and applications
• Mean value formulas on sublattices and flags of the random lattice
• Central Limit Theorems for Diophantine approximants