Binary Sequences Generated by Sequences {n}, n = 1, 2, . .

Abstrakt

In the paper arithmetical and pseudorandom properties of the set A of fractional parts of multiplies of an irrational number which belong to a given subinterval of the unit interval (0,1) are studied. It is proved here that the gaps between successive elements of A are at most of three lengths, a, b and a+b, which extends the known Slater's results to arbitrary intervals. From the other results, let us mention the exact description of the set of integers which are not equal to a difference of two arbitrary elements of A, or a new lower estimate of the Mauduit-Sarkozy well distribution measure, or a new proof of the mentioned Slater’s three gap theorems.