John J.C. Saunders, Department of Pure Mathematics, University of Waterloo
“Sums of Digits in q-ary expansions”
Let sq(n) denote the sum of the digits of a number n expressed in base q. We study here the ratio sq(nα)
sq
(n)
for
various
values
of
q
and
α.
In
1978,
Kenneth
B.
Stolarsky
showed
that
and that
lim inf s2(n2) = 0 n→∞ s2(n)
lim sup s2(n2) = ∞ n→∞ s2(n)
using an explicit construction. We show that for α = 2 and q ≥ 2, the above ratio can in fact be any positive rational number. If time permits, we also study what happens when α is a rational number that is not an integer, terminating the resulting expression by using the floor function.