Related papers: Properties of expansions in complex bases
Erd\H{o}s first introduced the idea of covering systems in 1950. Since then, much of the work in this area has concentrated on identifying covering systems that meet specific conditions on their moduli. Among the central open problems in…
Unique expansions in non-integer bases $q$ have been investigated in many papers during the last thirty years. They are often conveniently generated by labeled directed graphs. In the first part of this paper we give a precise description…
Given two real numbers $q_0,q_1$ with $q_0, q_1 > 1$ satisfying $q_0+q_1 \ge q_0q_1$, we call a sequence $(d_i)$ with $d_i\in \{0,1\}$ a $(q_0,q_1)$-expansion or a double-base expansion of a real number $x$ if \[…
In 2012, Diem introduced a new figure of merit for cryptographic sequences called expansion complexity. In this paper, we slightly modify this notion to obtain the so-called irreducible-expansion complexity which is more suitable for…
Expansions in noninteger bases often appear in number theory and probability theory, and they are closely connected to ergodic theory, measure theory and topology. For two-letter alphabets the golden ratio plays a special role: in smaller…
We consider \textit{additive spaces}, consisting of two intervals of unit length or two general probability measures on ${\mathbb R}^1$, positioned on the axes in ${\mathbb R}^2$, with a natural additive measure $\rho$. We study the…
Linear statistics of random zero sets are integrals of smooth differential forms over the zero set and as such are smooth analogues of the volume of the random zero set inside a fixed domain. We derive an asymptotic expansion for the…
A classical additive basis question is Waring's problem. It has been extended to integer polynomial and non-integer power sequences. In this paper, we will consider a wider class of functions, namely functions from a Hardy field, and show…
For a given positive integer $m$, let $A=\set{0,1,...,m}$ and $q \in (m,m+1)$. A sequence $(c_i)=c_1c_2 ...$ consisting of elements in $A$ is called an expansion of $x$ if $\sum_{i=1}^{\infty} c_i q^{-i}=x$. It is known that almost every…
This contribution is devoted to the study of positional numeration systems with negative base introduced by Ito and Sadahiro in 2009, called (-\beta)-expansions. We give an admissibility criterion for more general case of…
Simple extensions of peripheric extended twists, introduced recently by Lyakhovsky and Del Olmo, are presented. Explicit form of twisting elements are given and it is shown that the new twists as well as peripheric extended twists are…
We introduce a canonical isomorphism from the space of pure-type complex differential forms on a compact complex manifold to the one on its infinitesimal deformations. By use of this map, we generalize an extension formula in a recent work…
We show that for any fixed base $a$, a positive proportion of primes have the property that they become composite after altering any one of their digits in the base $a$ expansion; the case $a=2$ was already established by Cohen-Selfridge…
The notion of abundance of certain type of configuration in certain large sets was first proved by Furstenberg and Glazner in 1998. After that many author investigate abundance of different types of configurations in different types of…
Given a positive integer $M$ and a real number $q \in (1,M+1]$, an expansion of a real number $x \in \left[0,M/(q-1)\right]$ over the alphabet $A=\{0,1,\ldots,M\}$ is a sequence $(c_i) \in A^{\mathbb N}$ such that…
In additive number theory, a finite set $A$ of integers is an $h$-basis for $n$ if every integer in $\{0,1,2,\ldots, n\}$ can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. This paper introduces a new…
The main results of this paper are the construction, both rigourous and intuitive, of "the" intrinsic extension of the set of non negative integers N and the smallest over-field of R set which is continue (according to R.Dedekind). The aim…
A survey of recent results in elementary number theory is presented in this paper. Special attention is given to structure and asymptotic properties of certain families of positive integers.
A set of non-negative integers is an additive basis with range $n$, if its sumset covers all consecutive integers from 0 to $n$, but not $n+1$. If the range is exactly twice the largest element of the basis, the basis is restricted.…
An expansion upon Donald Kunth's quarter-imaginary base system is introduced to handle any imaginary number base where its real part is zero and the absolute value of its imaginary part is greater than one. A brief overview on number bases…