*Popis:* | Given n positive numbers a_1, ..., a_n in R consider all 2^n possible sums of them with signs + and -. How many of them can be equal? This is a variant of the so-called Littlewood-Offord problem, which was solved by Erdős in 1945 using Sperner's inequality. Since then the problem was considered in a more general setting when the numbers a_i are replaced by non-zero vectors in a R^k, the research culminating in a theorem by Kleitman (1970) for vectors a_i in arbitrary normed spaces. We consider a natural extension of the problem to the setting when a_i's are elements of a group: how many elements of the form g_1 * ... * g_n, where g_i is either a_i or a_i^{-1}, can coincide? Strengthening and generalizing some very recent results by Tiep and Vu, we establish several optimal inequalities, which, among other things, imply that the Erdős' bound for real numbers is still best possible for any torsion-free group. (Joint work with G. Šemetulskis). |