Related papers: Most numbers are not normal

We are interested in the set of normal sequences in the space $\{0,1\}^\mathbb{N}$ with a given frequency of the pattern $11$ in the positions $k, 2k$. The topological entropy of such sets is determined.

Fix a positive integer $N\geq2$. For a real number $x\in[0,1]$ and a digit $i\in\{0, 1,...,N-1\}$, let $\Pi_i(x, n)$ denote the frequency of the digit $i$ among the first $n$ $N$-adic digits of $x$. It is well-known that for a typical (in…

Fix a sequence of integers $Q=\{q_n\}_{n=1}^\infty$ such that $q_n$ is greater than or equal to 2 for all $n$. In this paper, we improve upon results by J. Galambos and F. Schweiger showing that almost every (in the sense of Lebesgue…

In this paper we recall a non-standard construction of the Borel sigma-algebra B in [0,1] and construct a family of measures (in particular, Lebesgue measure) in B by a completely non-topological method. This approach, that goes back to…

Let $S \subseteq \mathbb{N}$ have the property that for each $k \in S$ the set $(S - k) \cap \mathbb{N} \setminus S$ has asymptotic density $0$. We prove that there exists a basic sequence $Q$ where the set of numbers $Q$-normal of all…

We show that an ideal $\mathcal{I}$ on the positive integers is meager if and only if there exists a bounded nonconvergent real sequence $x$ such that the set of subsequences [resp. permutations] of $x$ which preserve the set of…

Let $\mathscr{X}$ be the set of positive real sequences $x=(x_n)$ such that the series $\sum_n x_n$ is divergent. For each $x \in \mathscr{X}$, let $\mathcal{I}_x$ be the collection of all $A\subseteq \mathbf{N}$ such that the subseries…

A $0,1$ matrix is said to be regular if all of its rows and columns have the same number of ones. We prove that for infinitely many integers $k$, there exists a square regular $0,1$ matrix with binary rank $k$, such that the Boolean rank of…

A real number $x$ is considered normal in an integer base $b \geq 2$ if its digit expansion in this base is ``equitable'', ensuring that for each $k \geq 1$, every ordered sequence of $k$ digits from $\{0, 1, \ldots, b-1\}$ occurs in the…

Let $x$ be a sequence taking values in a separable metric space and $\mathcal{I}$ be a generalized density ideal or an $F_\sigma$-ideal on the positive integers (in particular, $\mathcal{I}$ can be any Erd{\H o}s--Ulam ideal or any summable…

For any constant $C_0>0$, we construct a set $A \subset {\mathbb N}$ such that one has $$ \sum_{n \in A: n \leq x} \frac{1}{n} = \exp\left(\left(\frac{C_0}{2}+o(1)\right) (\log\log x)^{1/2} \log\log\log x \right)$$ and $$ \sum_{n,m \in A:…

We show that for every positive integer $k$ there are positive constants $C$ and $c$ such that if $A$ is a subset of $\{1, 2, \dots, n\}$ of size at least $C n^{1/k}$, then, for some $d \leq k-1$, the set of subset sums of $A$ contains a…

We give a construction of an absolutely normal real number $x$ such that for every integer $b $ greater than or equal to $2$, the discrepancy of the first $N$ terms of the sequence $(b^n x \mod 1)_{n\geq 0}$ is of asymptotic order…

We introduce an ergodic approach to the study of {\em joint normality} of representations of numbers. For example, we show that for any integer $b \geq 2$ almost every number $x \in [0,1)$ is jointly normal with respect to the $b$-expansion…

In this paper, we show that the concatenation of the Fibonacci sequence is \textit{normal} in base $10$, meaning every string of a given length, $k$, occurs as frequently as every other string of length $k$ (there are as many $1$'s as $2$'s…

Fix $\alpha \in (0,1/3)$. We show that, from a topological point of view, almost all sets $A\subseteq \mathbb{N}$ have the property that, if $A^\prime=A$ for all but $o(n^{\alpha})$ elements, then $A^\prime$ is not a nontrivial sumset…

We introduce and study a new topological notion of the size for subsets of the real line, called \emph{super-density}. A set $A\subset\mathbb{R}$ is super-dense if for every non-empty open interval $I$ and every nowhere constant continuous…

We show that if there exists an integer subject to some congruence conditions that cannot be written as the sum of the norm of an ideal in $\mathbb{Z}[\exp(2\pi i/2^k)]$ and at most $k$ powers of $2$, $k\geq 3$, then there are infinitely…

We consider sets of reals $X$ endowed with the Sorgenfrey lower limit topology denoted $X[\leq]$. Przymusi\'nski proved that if $X$ is a $Q$-set then $(X[\leq])^2$ is normal. While the converse is not in general true we consider examples of…

Let $X_1$, $X_2$,... be a sequence of independent random variables with common distribution function $F$ in the domain of attraction of a Gumbel extreme value distribution and for each integer $n\geq 1$, let $X_{1,n} \leq ... X_{n,n}$…