An unresolved analogue of the Littlewood Conjecture

Abstract

This article begins with an introduction to a conjecture made around 1930 in the area of Diophantine approximation: the Littlewood Conjecture. The conjecture asks whether any two real numbers can be simultaneously well approximated by rational numbers with the same denominator. The introduction also focuses briefly on an analogue of this conjecture, regarding power series and polynomials with coefficients in an infinite field. Harold Davenport and Donald Lewis disproved this analogue of the Littlewood Conjecture in 1963. Following the introduction we focus on a claim relating to another analogue of this conjecture. In 1970, John Armitage believed that he had disproved an analogue of the Littlewood Conjecture, regarding power series and polynomials with coefficients in a finite field. The remainder of this article shows that Armitage’s claim was false.