On The Equidistributed(mod1) Of Real Sequences

Abstract

A sequence (an) of real numbers is equidistributed on an interval if the probability of finding any terms in any subinterval is proportional to the length of the subinterval. And is said to be equidistributed modulo 1 or uniformly distributed modulo 1 if the sequence of the fractional parts of an, denoted by (an) , is equidistributed in the interval [0; 1]. For any given real numbers r 0 and > 0 Koksma and H. Weyl proved respectively that the setsE of all positive real numbers r 0 and the sets Wr of all positive real numbers > 0, for which the sequence rn n2N is not uniformly distributed modulo 1, have Lebesgue measure zero. In this memoir, we give some algebraic properties of certain sets E a and show, among other things, that the sets Wr are uncountable.