Sums of products of fractional parts

TL;DR

This paper establishes bounds for sums involving fractional parts using Fourier analysis, introduces strongly badly approximable matrices, and proves a transference principle related to Diophantine approximation.

Contribution

It provides new bounds for sums of fractional parts, introduces the concept of strongly badly approximable matrices, and proves a transference principle connecting these matrices to Diophantine approximation.

Findings

Upper bounds are sharp in certain cases with counterexamples to Littlewood's conjecture.

Introduces strongly badly approximable matrices as a generalization.

Proves a transference principle for these matrices.

Abstract

We prove upper and lower bounds for certain sums of products of fractional parts by using majoring and minorizing functions from Fourier analysis. In special cases the upper bounds are sharp if there exist counterexamples to the Littlewood conjecture in Diophantine approximation. We introduce a generalization of such counterexamples which we call strongly badly approximable matrices. And we prove a transference principle for strongly badly approximable matrices.