Related papers: Ternary expansions of powers of 2

Motivated by a question of Erd\H{o}s, this paper considers questions concerning the discrete dynamical system on the 3-adic integers given by multiplication by 2. Let the 3-adic Cantor set consist of all 3-adic integers whose expansions use…

The \textit{ternary digits of $2^n$} are a finite sequence of 0s, 1s, and 2s. It is a natural question to ask whether the frequency of any string of 0s, 1s, and 2s in this sequence approaches the same limit for all strings of the same…

Erd\H{o}s conjectured that 1, 4, and 256 are the only powers of two whose ternary representations consist solely of 0s and 1s. Sloane conjectured that, except for $\{2^0,2^1,2^2,2^3,2^4,2^{15}\}$, every other power of two has at least one 0…

A two-dimensional string is simply a two-dimensional array. We continue the study of the combinatorial properties of repetitions in such strings over the binary alphabet, namely the number of distinct tandems, distinct quartics, and runs.…

This short note studies the asymptotic behavior of a generating function associated with the decimal expansion of \(2^n\). Our aims are twofold: (i) to present a problem on the best possible upper bound for this behavior, and (ii) to…

In a previous paper an asymptotic expansion for lambda_d in powers of 1/d was developed. The results of computer computations for some terms in the expansion, as well as various quantities associated to the expansion, are herein presented.…

On the set of positive integers, we consider the iterative process that maps $n$ to either $\frac{3n+1}{2}$ or $\frac{n}{2}$ depending on the parity of $n$. The Collatz conjecture states that all such sequences eventually enter the trivial…

In DOI:10.1017/etds.2022.2 the author proved that for each integer $k$ there is an implicit number $M > 0$ such that if $b_1, \cdots , b_k$ are multiplicatively independent integers greater than $M$, there are infinitely many integers whose…

In a landmark paper on arithmetical properties of Lambert series, Erd\H{o}s proved that $\sum_{n=1}^{\infty} \frac{1}{2^{n} - 1}$ is irrational. This value $E$ is now referred to as the Erd\H{o}s-Borwein constant. Crandall, in 2012, studied…

In this paper we examine the possibility of describing omitting types 1 and 2 by two at most ternary terms and any number of linear identities. All possible cases of systems of linear identities on two at most ternary terms are being…

We consider infinite iterated function systems $\{f_i\}_{i=1}^{\infty}$ on $[0,1]$ with a polynomially increasing contraction rate. We look at subsets of such systems where we only allow iterates $f_{i_1}\circ f_{i_2}\circ f_{i_3}\circ...$…

For $\lambda \in (1/2, 1)$ and $\alpha$, we consider sets of numbers $x$ such that for infinitely many $n$, $x$ is $2^{-\alpha n}$-close to some $\sum_{i=1}^n \omega_i \lambda^i$, where $\omega_i \in \{0,1\}$. These sets are in Falconer's…

For an infinite iterated function system $\mathbf{f}$ on $[0,1]$ with an attractor $\Lambda(\mathbf{f})$ and for an infinite subset $D\subseteq \mathbb{N}$, consider the set \[ \mathbb E(\mathbf{f},D)= \{ x \in \Lambda(\mathbf{f}):…

We establish sharp bounds for the Hausdorff dimension of sets of irrational numbers in $(0,1)$ whose digits in the $N$-expansion are either uniformly bounded or tend to infinity. For sets with digits bounded by an integer $M \ge N$, we…

This paper continues the study of the structure of finite intersections of general multiplicative translates $\mathcal{C}(M_1,\ldots,M_n)=\frac{1}{M_1}\Sigma_{3,\bar{2}}\cap\cdots\cap\frac{1}{M_n}\Sigma_{3,\bar{2}}$ for integers $1\leq…

We show that the number of $1$'s in the first $N$ digits of the binary expansion of $\sqrt{2}$ is at least $\sqrt{2N}(1+o(1))$ and show that this bound can be improved to around $2\sqrt{N}/\sqrt{2\sqrt{2}-1}$ infinitely often.

A folklore conjecture in number theory states that the only integers whose expansions in base $3,4$ and $5$ contain solely binary digits are $0, 1$ and $82000$. In this paper, we present the first progress on this conjecture. Furthermore,…

The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when…

Let $\alpha, \beta$ be two relatively prime algebraic integers in a number field $K$ and $N$ be a positive integer. We show that the number of $n\in\{1,2,\dots,N\}$ such that the $\beta$-adic expansion of $\alpha^n$ omits a given digit is…

Motivated by Erd\H{o}s' ternary conjecture and by recent work of Cui--Ma--Jiang [``Geometric progressions meet Cantor sets'', \textit{Chaos Solitons Fractals} \textbf{163} (2022), 112567.] on intersections between geometric progressions and…