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A set $A\subseteq\newcommand{\N}{\mathbb{N}}\N$ is said to be Golomb if whenever $a<b \in A$ and $a'<b' \in A$ with $(b-a) = (b' - a')$, then $a=a'$ and $b=b'$. If $A\subseteq \N$ is Golomb, we let $\ …

For any set $X$, let $\newcommand{\N}{\mathbb{N}}[X]^2 = \big\{\{x,y\}:x\neq y \in X\big\}$ and set $[n]^2 = [\{0,\dotsc,n-1\}]^2$ for any positive integer $n$. For $A\subseteq [\N]^2$ we set $$\newco …

We call a collection ${\cal S}\subseteq {\cal P}(\newcommand{\N}{\mathbb{N}}\N)$ bijection-proof if for any bijection $\varphi:\N\to\N$ there is $T\in{\cal S}$ with $\varphi(T) \in {\cal S}$. For any …

Motivation. If we consider any bijection $\varphi:\newcommand{\N}{\mathbb{N}} \N \to \N$, we say integers $m\neq n$ are shrinking with respect to $\varphi$ if $|m-n|>|\varphi(m) - \varphi(n)|$, and ex …

The following question is loosely based on the friendship paradox. Let $G=(V,E)$ be a simple, undirected graph. For $v\in V$, we let the neighborhood of $v$ be $N(v) = \big\{w\in V:\{v,w\}\in E\big\}$ …

Let $\mathbb{N} = \{0,1,2,\ldots\}$ denote the set of non-negative integers. If $n\in\mathbb{N}$ we let $[n] = \{0,\ldots,n-1\}$. For $A \subseteq \mathbb{N}$ we let $$\mu^+(A) = \lim\sup_{n\to\infty} …

Donald Knuth suggested a bitwise approximation for addition on the non-negative integers that is very fast on common processors: $(a,b)\mapsto (a\oplus b) \oplus ((a\land b) \ll 1)$, where $a,b$ are g …

For any $A\subseteq \mathbb{N}$ we let the (lower) density of $A$ be defined by $$d(A) = \liminf_{n\to\infty}\frac{|A\cap\{0,\ldots,n\}|}{n+1}.$$ If $\pi:\mathbb{N}\to\mathbb{N}$ is a permutation (bij …

Let $F$ be the set of all integers $n>1$ such that in the Fibonacci sequence modulo $n$, the value $0$ occurs infinitely often. What is the value of $\lim\sup_{n\to\infty}\frac{|F\cap\{0,\ldots,n\}|}{ …

For $A\subseteq \omega$ we let the lower and upper density be defined as $$\mu^-(A):= \lim\inf_{n\to\infty}\frac{|A\cap n|}{n+1} \text{ and } \mu^+(A):= \lim\sup_{n\to\infty}\frac{|A\cap n|}{n+1}$$ re …

Let $\mathbb{N}_+$ denote the set of positive integers and let $\mathbb{N}_0 = \mathbb{N}_+\cup\{0\}$. Fix $\alpha\in[0,1]\setminus \mathbb{Q}$. For $n\in\mathbb{N}_+$ we let the approximation radius …

A positive irrational number $\alpha\in{\mathbb R}\setminus {\mathbb Q}$ is said to be a Brjuno number if $$\sum_{i=1}^\infty\frac{\log q_{i+1}}{q_i} < \infty$$ where $q_i>0$ is the denominator of the …

Let $\mathbb{N}_+$ denote the set of positive integers. Consider the remainder function $\text{rem}:\mathbb{N}_+\times \mathbb{N}_+ \to \mathbb{N}\cup\{0\}$ defined by $$(n,d) \mapsto n - \Big(\Big\lf …

Does the set of squares $S = \{n^2: n\in\omega\}$ adhere to Benford's law for the first digit in every base $b\geq 2$? Precise formulation of what it means for a set $T\subseteq \omega$ to "adhere to …

We say that a set of natural numbers $A\subseteq \omega$ has positive upper density if $$\lim\sup_{n\to\infty}\frac{|A\cap n|}{n+1} > 0.$$ Szeméredi's theorem states that every $A\subseteq \omega$ hav …