  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.
