Stolarsky’s conjecture and the sum of digits of polynomial values

Abstract:

Let $s_q(n)$ denote the sum of the digits in the $q$-ary expansion of an integer $n$. In 1978, Stolarsky showed that $\displaystyle { \liminf _{n\to \infty }} \frac {s_2(n^2)}{s_2(n)} = 0$. He conjectured that, just as for $n^2$, this limit infimum should be 0 for higher powers of $n$. We prove and generalize this conjecture showing that for any polynomial $p(x)=a_h x^h+a_{h-1} x^{h-1} + \dots + a_0 \in \mathbb {Z}[x]$ with $h\geq 2$ and $a_h>0$ and any base $q$, \[ \liminf _{n\to \infty } \frac {s_q(p(n))}{s_q(n)}=0.\] For any $\varepsilon > 0$ we give a bound on the minimal $n$ such that the ratio $s_q(p(n))/ s_q(n) < \varepsilon$. Further, we give lower bounds for the number of $n < N$ such that $s_q(p(n))/s_q(n) < \varepsilon$.