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 $$\liminf_{n\to\infty} \frac{s_2(n^2)}{s_2(n)} = 0.$$ 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 $a_h>0$ and any base $q$, $$\liminf_{n\to\infty} \frac{s_q(p(n))}{s_q(n)}=0.$$ For any $\epsilon > 0$ we give a bound on the minimal $n$ such that the ratio $s_q(p(n))/s_q(n) < \epsilon$. Further, we give lower bounds for the number of $n < N$ such that $s_q(p(n))/s_q(n) < \epsilon$.Comment: 13 page

The sum of digits of nn and n2n^2

Let $s_q(n)$ denote the sum of the digits in the $q$-ary expansion of an integer $n$. In 2005, Melfi examined the structure of $n$ such that $s_2(n) = s_2(n^2)$. We extend this study to the more general case of generic $q$ and polynomials $p(n)$, and obtain, in particular, a refinement of Melfi's result. We also give a more detailed analysis of the special case $p(n) = n^2$, looking at the subsets of $n$ where $s_q(n) = s_q(n^2) = k$ for fixed $k$.Comment: 16 page

Thue-Morse at Multiples of an Integer

Let (t_n) be the classical Thue-Morse sequence defined by t_n = s_2(n) (mod 2), where s_2 is the sum of the bits in the binary representation of n. It is well known that for any integer k>=1 the frequency of the letter "1" in the subsequence t_0, t_k, t_{2k}, ... is asymptotically 1/2. Here we prove that for any k there is a n<=k+4 such that t_{kn}=1. Moreover, we show that n can be chosen to have Hamming weight <=3. This is best in a twofold sense. First, there are infinitely many k such that t_{kn}=1 implies that n has Hamming weight >=3. Second, we characterize all k where the minimal n equals k, k+1, k+2, k+3, or k+4. Finally, we present some results and conjectures for the generalized problem, where s_2 is replaced by s_b for an arbitrary base b>=2.Comment: 14 page

On a conjecture of Dekking : The sum of digits of even numbers

Let $q\geq 2$ and denote by $s_q$ the sum-of-digits function in base $q$. For $j=0,1,...,q-1$ consider # \{0 \le n < N : \;\;s_q(2n) \equiv j \pmod q \}. In 1983, F. M. Dekking conjectured that this quantity is greater than $N/q$ and, respectively, less than $N/q$ for infinitely many $N$, thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.Comment: 6 pages, accepted by JTN