On the exponential sum with the sum of digits of hereditary base b notation

Let b 2 be an integer and wb(n) be the sum of digits of the nonnegative integer n written in hereditary base b notation. We give optimal upper bounds for the exponential sum PN1 n=0 exp(2⇡iwb(n)t), where t is a real number. In particular, our results imply that for each positive integer m the sequence {wb(n)}1 n=0 is uniformly distributed modulo m; and that for each irrational real ↵ the sequence {wb(n)↵}1 n=1 is uniformly distributed modulo 1.