Related papers: Ternary expansions of powers of 2

Motivated by works on extension sets in standard domains we introduce a notion of the Carath\'eodory set that seems better suited for the methods used in proofs of results on characterization of extension sets. A special stress is put on a…

We introduce a family of maps generating continued fractions where the digit $1$ in the numerator is replaced cyclically by some given non-negative integers $(N_1,\ldots,N_m)$. We prove the convergence of the given algorithm, and study the…

The one-dimensional Dirac operator \begin{equation*} L = i \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \frac{d}{dx} +\begin{pmatrix} 0 & P(x) \\ Q(x) & 0 \end{pmatrix}, \quad P,Q \in L^2 ([0,\pi]), \end{equation*} considered on $[0,\pi]$…

In this paper, we study linear forms \[\lambda = \beta_1\mathrm{e}^{\alpha_1}+\cdots+\beta_m\mathrm{e}^{\alpha_m},\] where $\alpha_i$ and $\beta_i$ are algebraic numbers. An explicit lower bound for the absolute value of $\lambda$ is…

We consider equally-weighted Cantor measures $\mu_{q,b}$ arising from iterated function systems of the form ${b^{-1}(x+i)}$, $i=0,1,...,q-1$, where $q<b$. We classify the $(q,b)$ so that they have infinitely many mutually orthogonal…

We obtain the following results about the avoidance of ternary formulas. Up to renaming of the letters, the only infinite ternary words avoiding the formula $ABCAB.ABCBA.ACB.BAC$ (resp. $ABCA.BCAB.BCB.CBA$) have the same set of recurrent…

Let $$\lambda(s)=\sum_{n=0}^\infty\frac1{(2n+1)^s},$$ $$\beta(s)=\sum_{n=0}^\infty\frac{(-1)^{n}}{(2n+1)^s},$$ and $$\eta(s)=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s}$$ be the Dirichlet lambda function, its alternating form, and the Dirichlet…

Berry and Tabor conjectured in 1977 that spectra of generic integrable quantum systems have the same local statistics as a Poisson point process. We verify their conjecture in the case of the two-point spectral density for a quantum…

We have primarily obtained three results on numbers of the form $p + 2^k$. Firstly, we have constructed many arithmetic progressions, each of which does not contain numbers of the form $p + 2^k$, disproving a conjecture by Erd\H{o}s as Chen…

The authors have recently obtained a lower bound of the Hausdorff dimension of the sets of vectors $(x_1, \ldots, x_d)\in [0,1)^d$ with large Weyl sums, namely of vectors for which $$ \left| \sum_{n=1}^{N}\exp(2\pi i (x_1 n+\ldots +x_d…

For $q>1$ we consider expansions in base $q$ over the alphabet $\{0,1,q\}$. Let $\mathcal{U}_q$ be the set of $x$ which have a unique $q$-expansions. For $k=2, 3,\cdots,\aleph_0$ let $\mathcal{B}_k$ be the set of bases $q$ for which there…

Deterministic and probabilistic communication protocols are introduced in which parties can exchange the values of polynomials (rather than bits in the usual setting). It is established a sharp lower bound $2n$ on the communication…

In a paper entitled Binary lambda calculus and combinatory logic, John Tromp presents a simple way of encoding lambda calculus terms as binary sequences. In what follows, we study the numbers of binary strings of a given size that represent…

Let $\theta(n)$ denote the number of permutations of $\{1,2,\ldots,n\}$ that do not contain a 3-term arithmetic progression as a subsequence. Such permutations are known as 3-free permutations. We present a dynamic programming algorithm to…

We consider a couple of versions of classical Kurosh problem (whether there is an infinite-dimensional algebraic algebra?) for varieties of linear multioperator algebras over a field. We show that, given an arbitrary signature, there is a…

Using discretized orthogonal systems (curvature line systems) with periodicity, created using Darboux transformations and their permutability, we have discrete and semi-discrete k-dimensional isothermic tori which are full in n-dimensional…

The recently confirmed Dejean's conjecture about the threshold between avoidable and unavoidable powers of words gave rise to interesting and challenging problems on the structure and growth of threshold words. Over any finite alphabet with…

The upper bound for asymptotic behavior of the coefficients of expansion of the evolution operator kernel in powers of the time interval $\Dt$ was obtained. It is found that for the nonpolynomial potentials the coefficients may increase as…

In this paper analyzes \textit{The Erd\H{o}s-Straus conjecture} asserts that $f$$(n)$ $>$ 0 for every $n$ $\geq$ 2, where $f(n)$ indicates the number of solutions to the Diophantine Equation…

Motivated by algorithmic problems from combinatorial group theory we study computational properties of integers equipped with binary operations +, -, z = x 2^y, z = x 2^{-y} (the former two are partial) and predicates < and =. Notice that…