A solution to a problem of Cassels and Diophantine properties of cubic numbers

TL;DR

This paper proves that almost all pairs of real numbers satisfy a uniform inhomogeneous version of Littlewood's conjecture, solving a long-standing problem of Cassels and exploring Diophantine properties of cubic numbers.

Contribution

It establishes that almost any pair of real numbers satisfies the inhomogeneous Littlewood's conjecture, and identifies conditions under which specific algebraic triples do or do not satisfy it.

Findings

Almost all pairs (a,b) satisfy the conjecture.

Pairs (a,b) from totally real number fields satisfy the conjecture.

Linearly dependent triples (1,a,b) do not satisfy the conjecture.

Abstract

We prove that almost any pair of real numbers a,b, satisfies the following inhomogeneous uniform version of Littlewood's conjecture: (*) forall x,y in R, liminf_{|n|\to\infty} |n|<na - x> <nb - y> = 0, where <-> denotes the distance from the nearest integer. The existence of even a single pair that satisfies (*), solves a problem of Cassels from the 50's. We then prove that if 1,a,b span a totally real number field, then a,b, satisfy (*). It is further shown that if 1,a,b, are linearly dependent over Q, a,b cannot satisfy (*). The results are then applied to give examples of irregular orbit closures of the diagonal groups of a new type.