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

Two-dimensional self-affine sets with interior points, and the set of uniqueness

Let $M$ be a $2\times2$ real matrix with both eigenvalues less than~1 in modulus. Consider two self-affine contraction maps from $\mathbb R^2 \to \mathbb R^2$, \begin{equation*} T_m(v) = M v - u \ \ \mathrm{and}\ \ T_p(v) = M v + u, \end{equation*} where $u\neq0$. We are interested in the properties of the attractor of the iterated function system (IFS) generated by $T_m$ and $T_p$, i.e., the unique non-empty compact set $A$ such that $A = T_m(A) \cup T_p(A)$. Our two main results are as follows: 1. If both eigenvalues of $M$ are between $2^{-1/4}\approx 0.8409$ and $1$ in absolute value, and the IFS is non-degenerate, then $A$ has non-empty interior. 2. For almost all non-degenerate IFS, the set of points which have a unique address is of positive Hausdorff dimension -- with the exceptional cases fully described as well. This paper continues our work begun in [11].Comment: 29 pages, 7 figure

Multidimensional self-affine sets: non-empty interior and the set of uniqueness

Let $M$ be a $d\times d$ contracting matrix. In this paper we consider the self-affine iterated function system $\{Mv-u, Mv+u\}$, where $u$ is a cyclic vector. Our main result is as follows: if $|\det M|\ge 2^{-1/d}$, then the attractor $A_M$ has non-empty interior. We also consider the set $\mathcal U_M$ of points in $A_M$ which have a unique address. We show that unless $M$ belongs to a very special (non-generic) class, the Hausdorff dimension of $\mathcal U_M$ is positive. For this special class the full description of $\mathcal U_M$ is given as well. This paper continues our work begun in two previous papers.Comment: 10 pages, no figure

A lower bound for the dimension of Bernoulli convolutions

Let $\beta\in(1,2)$ and let $H_\beta$ denote Garsia's entropy for the Bernoulli convolution $\mu_\beta$ associated with $\beta$. In the present paper we show that $H_\beta>0.82$ for all $\beta \in (1, 2)$ and improve this bound for certain ranges. Combined with recent results by Hochman and Breuillard-Varj\'u, this yields $\dim (\mu_\beta)\ge0.82$ for all $\beta\in(1,2)$. In addition, we show that if an algebraic $\beta$ is such that $[\mathbb{Q}(\beta): \mathbb{Q}(\beta^k)] = k$ for some $k \geq 2$, then $\dim(\mu_\beta)=1$. Such is, for instance, any root of a Pisot number which is not a Pisot number itself.Comment: 8 pages, no figure

Open maps: small and large holes with unusual properties

Let $X$ be a two-sided subshift on a finite alphabet endowed with a mixing probability measure which is positive on all cylinders in $X$. We show that there exist arbitrarily small finite overlapping union of shifted cylinders which intersect every orbit under the shift map. We also show that for any proper subshift $Y$ of $X$ there exists a finite overlapping unions of shifted cylinders such that its survivor set contains $Y$ (in particular, it can have entropy arbitrarily close to the entropy of $X$). Both results may be seen as somewhat counter-intuitive. Finally, we apply these results to a certain class of hyperbolic algebraic automorphisms of a torus.Comment: 15 pages, no figure