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 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.