ENTROPY AND UNIFORM DISTRIBUTION OF ORBITS IN Td

We present a class of integer sequences fc n g with the property that for every p-invariant and ergodic positive-entropy measure on T, fc n x (mod 1)g is uniformly distributed for-almost every x. This extends a result of B. Host, who proved this for the sequence fq n g, for q relatively prime to p. Our class of sequences includes, for instance, the sequence c n = P f i (n)q n i , where the numbers q i are distinct and are relatively prime to p and f i are any polynomials. More generally, recursion sequences for which the free coee-cient of the recursion polynomial is relatively prime to p are in this class as well, provided they satisfy a simple irreducibility condition. In the multi-dimensional case we derive suucient conditions for a pair of endomorphisms A; B 2 End(T d) (with A diagonal) and an A-invariant and ergodic measure , such that B-orbits of the form fB n !g are uniformly distributed for-almost every ! 2 T d .