On the distribution of fractions with power denominator

2019/01/07 by Bryce Kerr, Kerr, Bryce
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1901.01687

openalex publication_date 2019/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we obtain a sharp upper bound for the number of solutions to a certain diophantine inequality involving fractions with power denominator. This problem is motivated by a conjecture of Zhao concerning the spacing of such fractions in short intervals and the large sieve for power modulus. As applications of our estimate we show Zhao's conjecture is true except for a set of small measure and give a new ℓ₁ → ℓ₂ large sieve inequality for power modulus.

Related

• The large sieve with power moduli for ℤ[i]
• On the distribution of partial quotients of reduced fractions with fixed denominator
• Moments of restricted divisor functions
• Large sieve inequality for power moduli
• Egyptian Fractions with odd denominators