Mathematics

Let alpha and beta be real numbers. We prove estimates for partial sums that are
sensitive to the distance from the integer multiples alpha n and beta n to the nearest integer.
We give both upper and lower bounds for such sums. It turns out that the lower bounds are
of the correct order of magnitude whenever the pair alpha, beta is a counterexample to a notorious
conjecture of Littlewood in Diophantine approximation. We also consider more general sums involving
products of many linear forms in many variables.

This is joint work with Thai Hoang Le.