Digital Expansion of Exponential Sequences

Abstract. We consider the q-ary digital expansion of the first N terms of an exponential sequence a. Using a result due to Kiss und Tichy [8], we prove that the average number of occurrences of an arbitrary digital block in the last c logN digits is asymptotically equal to the expected value. Under stronger assumptions we get a similar result for the first (logN) 3 2 −2 digits, where 2 is a positive constant. In both methods, we use estimations of exponential sums and the concept of discrepancy of real sequences modulo 1 plays an important role.