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

Let $s_q(n)$ denote the sum of the digits in the $q$-ary expansion of an integer $n$. In 1978, Stolarsky showed that $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{s_2(n^2)}{s_2(n)}=0.$$ He conjectured that, 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} + …

• a_0 \in \Z[x]$with$h\geq 2$and$a_h>0$andanybase$q$, $$\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{s_q(p(n))}{s_q(n)}=0.$$ For any $ϵ>0$ we give a bound on the minimal $n$ such that the ratio $s_q(p(n))/s_q(n) < \epsilon$\mathrm{.}Further,wegivelowerboundsforthenumberof$n < N$suchthat$s_q(p(n))/s_q(n) < \epsilon$.