Counting multiplicative approximations

A famous conjecture of Littlewood (c. 1930) concerns approximating two real numbers by rationals of the same denominator, multiplying the errors. In a lesser-known paper, Wang and Yu (Chin Ann Math 2:1–12, 1981) established an asymptotic formula for the number of such approximations, valid almost al...

Full description

Saved in:
Bibliographic Details
Published in:The Ramanujan journal 2023-09, Vol.62 (1), p.241-250
Main Authors: Chow, Sam, Technau, Niclas
Format: Article
Language:English
Subjects:
Approximation
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Lower bounds
Mathematics
Mathematics and Statistics
Number Theory
Real numbers
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:A famous conjecture of Littlewood (c. 1930) concerns approximating two real numbers by rationals of the same denominator, multiplying the errors. In a lesser-known paper, Wang and Yu (Chin Ann Math 2:1–12, 1981) established an asymptotic formula for the number of such approximations, valid almost always. Using the quantitative Koukoulopoulos–Maynard theorem of Aistleitner–Borda–Hauke, together with bounds arising from the theory of Bohr sets, we deduce lower bounds of the expected order of magnitude for inhomogeneous and fibre refinements of the problem.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-022-00610-3