sums of digits of powers of integers

It is known (Senge and Straus, 1971, see also C.L.Stewart, 1980) that for every natural $$a$$, not a power of 10, and every natural $$s$$, there are only finitely many $$k$$ such that the sum of decimal digits of $$a^k$$ does not exceed $$s$$. So let $$f(s)$$ be the largest $$k$$ with this property. What is the growth rate of $$f$$ ? In particular, is it always at most linear?

Update 1: Conjecture. $$\liminf S_{10}(a^k)/k > \log_2(a)$$. This would imply that $$f$$ is bounded by a linear function. Here $$S_{10}(u)$$ is the sum of decimal digits of $$u$$.

Update 2: As far as I know the best proved estimate of $$f$$ is double exponential (C.L. Stewart).

Update 3: Since somebody erased my comments below, I add it here. The problem is related to the exponential Diophantine equation $$10^{k_1}+...+10^{k_s}=a^k$$. See the book Shorey, T. N.; Tijdeman, R. Exponential Diophantine equations. Cambridge Tracts in Mathematics, 87. Cambridge University Press, Cambridge, 1986.

Update 4: Possibly a better conjecture than in Update 1. The distribution of digits in $$a^k$$ should be very close to uniform. Hence $$\liminf S_{10}(a^k)/k\approx 4.5\log_{10} a$$.

• @Mark: Just for my own education, could you identify in which Shorey-Tijdeman paper this is established? They have written quite a few papers together... – Joseph O'Rourke Sep 16 '10 at 16:05
• By the way, for $a=2$ or $5$, there is an old result of A. Schinzel, which is included in Sierpiński's "250 problems in elementary number theory". If I remember the proof correctly, this result shows that in this case $f$ grows at most linearly. – Mark Sapir Sep 16 '10 at 16:52
• @Joseph: I changed the references. – Mark Sapir Sep 18 '10 at 16:34
• The references in full: C L Stewart, On the representation of an integer in two different bases, J Reine Angew Math 319 (1980) 63-72, MR 81j:10012; H G Senge, E G Straus, PV-numbers and sets of multiplicity, Proceedings of the Washington State University Conference on Number Theory (1971) 55-67, MR 47 #8452. – Gerry Myerson Sep 20 '10 at 1:07
• @Gerry: Yes, it is correct according to MathSci. Also you may find in Stewart's paper a double exponential estimate for $f$. I have not been able to find better estimates in the literature although there exist much more recent papers on the subject generalizing Stewart's results, in particular. Calculations suggest a linear bound. – Mark Sapir Sep 20 '10 at 2:28