On distribution of integers with missing digits in residue classes
by
Sergei Konyagin
Moscow State University

Let

g in \bold N,     g >= 3,     t in \bold N,     2 <= t <= g-1, \tag1

D subset {0, 1, ..., g-1},     0 in D,    |D|=t, \tag2

and let W_D(N) denote the set of integers n such that 0 <= n <= N and representing n in the number system to base g:

n= \nu å j=0  ajgj,     0 <= aj <= g-1,

where g^\nu <= N < g^\nu+1, we have a_j in D for j=0, ..., \nu. P. Erdös, C. Mauduit and A. Sárközy proved, among many other interesting results, that the set W_D(N) is well-distributed in the modulo m residue classes if m < exp(c(g, t)(logN)^1/2). Also, they showed that if t > g^3/4, a positive integer z is small enough in terms of t and g (namely, z < (2(1-logt/logg))^-1), then W_D(N) contains integers with zth power parts as large as cN^c with c=c(g, t, z) > 0. It turns out that the last property holds for all g, t and z.

Date received: April 12, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacu-06.