105 problems

Let $\alpha\neq 0$ and let $\theta>0$ satisfy $\theta\neq\tfrac12$ and $\theta\notin\mathbb Z$. For $s>0$, define the pair-correlation statistic by … where $\|x\|$ denotes the dist…

Let $\alpha$ be a real number whose Diophantine type is defined by the existence of a constant $c_\alpha>0$ such that … for all coprime integers $p$ and $q$. Consider the sequence…

Equality conjecture.

Let $\alpha$ be a badly approximable real number, meaning that there exists a constant $c>0$ such that … for all rational numbers $p/q$. For a fixed integer $d\geq 2$, consider the…

The Fibonacci set is a point set associated with the Fibonacci permutation, considered as a construction on the two-dimensional torus $\mathbb{T}^2$. Fibonacci-set conjecture. The…

Let $t(n)$ denote the Thue–Morse sequence, and let $P(x)$ be a non-negative integer-valued polynomial. Gelfond's conjecture. The values of $t(P(n))$ are equidistributed between $0$…

Let $p$ be prime, let $n=1$, and let $\vartheta=2$ be a primitive root modulo $p$. Consider the consecutive powers $\vartheta^m$ and $\vartheta^{m+1}$ modulo $p$. Arnold's independ…

Let $p$ be prime, let $n$ be a positive integer, and let $\mathbb{F}_{p^n}$ be the finite field with $p^n$ elements. Fix a primitive root $\vartheta$ of $\mathbb{F}_{p^n}$ and writ…

Khintchine's strong uniform distribution conjecture. For every $f\in L^1([0,1))$, the averages

Let $R(g,n)$ denote the maximum size of a $B^{\ast}[g]$ set contained in $\{1,\dots,n\}$. A sequence $(S_n)_{n=1}^\infty$ of sets of positive integers becomes uniformly distributed…

Let $F_N(r)$ denote the empirical distribution of $q_{\min}\bmod \ell$, let $\pi_{\mathrm{perm}}(r)$ be the permitted-residue distribution, and let $C_{\mathrm{opt}}(k,\ell)$ be a…

Let $d\geq 2$, and define functions on $[0,1]^d$ by … … and … These functions aggregate only distances between consecutive arguments, unlike the pairwise-distance functions conside…

Let $(\omega_n)$ be a $p$-Bernoulli sequence of $2$ or $3$, meaning that the $\omega_n$ are independent and satisfy $P(\omega_n=2)=p=1-P(\omega_n=3)$ for fixed $0<p<1$. Consider th…

Let $E=(\lambda_n)_{n\ge 1}\subset\mathbb Z$ be a sequence of distinct integers, let $\mathbb T=\mathbb R/\mathbb Z$ be the circle with Lebesgue measure $\mathbf m$, and let $f$ be…

For each positive integer $n$, write the ternary expansion of $2^n$ as a finite string. Individual-power digit-frequency conjecture. In the ternary expansion of each individual pow…

For each positive integer $n$, let $S_n$ be the base-$3$ expansion of $2^n$. For a fixed positive integer $k$, consider the aggregate frequencies of the length-$k$ strings occurrin…

For each positive integer $n$, write the base-$3$ expansion of $2^n$ as a finite string $S_n$, and let $F_d(N)$ be the aggregate frequency of the digit $d\in\{0,1,2\}$ among the st…

Let $g_1,\dots,g_k:\mathbb{R}\to(1,\infty)$ and $f_1,\dots,f_k:\mathbb{R}\to\mathbb{R}$ be analytic functions. Let $a_1,\dots,a_k,b_1,\dots,b_k:\mathbb{N}\to\mathbb{R}$ be sequence…

Let $I\subset\mathbb{R}$ be a compact interval and let $k\in\mathbb{N}$. For each $i=1,\dots,k$, let $(u_{i,n})_{n\in\mathbb{N}}$ be a sequence of functions in $C^1(I,\mathbb{R})$…

Let $(u_n)_{n\in\mathbb{N}}$ satisfy the conditions of the cited Koksma theorem, and let $f_1,\dots,f_k:\mathbb{R}\to\mathbb{R}$ be analytic functions such that $1,f_1,\dots,f_k$ a…

Let $g:\mathbb{R}\to(1,\infty)$ be a nonconstant analytic function, let $b:\mathbb{N}\to\mathbb{R}$ tend to infinity and be scattered, let $k\in\mathbb{N}$, let…

Let $n\in\mathbb N$, let $\omega=(\boldsymbol{\omega}_j)_{j\geq 1}$ be a sequence in $[0,1]^n$, and let $\Psi=(\psi_1,\ldots,\psi_n)$ be an $n$-tuple of non-increasing functions. W…

Larcher–Stockinger conjecture. If for almost all $\alpha$ the pair correlations of $(a_n\alpha)_{n \in \mathbb{N}} \mod 1$ are not Poissonian, then the pair correlations of this se…

Rational accumulation-point conjecture. Every rational number $r$ with $0\leq r\leq 1$ is an accumulation point of this sequence. This conjecture concerns the distribution modulo o…

Let $(x_n)_{n\geq 1}$ be a sequence in $[0,1)^d$, and let … where the supremum is over all intervals $B=[0,b_1)\times\cdots\times[0,b_d)$ with $0\leq b_i\leq 1$. Optimal-order conj…