Sums of products of fractional parts

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 lower bounds are sharp if there exist counterexamples to the Littlewood conjecture in Diophantine approximation. We show that our more general lower bounds are sharp if there exist multiplicatively badly approximable matrices as considered in recent papers of Y. Bugeaud and O. N. German. We prove upper bounds for sums of products of fractional parts that hold on the complement of a set of small measure. We also give an alternative proof of German's transference principle for multiplicatively badly approximable matrices.