34 problems

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 $I\subset\mathbb{R}$ 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…

Uniform-distribution conjecture. As $p\to\infty$,

Uniform distribution conjecture. The $d$-dimensional interlaced Halton sequence in Algorithm is uniformly distributed.

Monotonicity conjecture. If $x$ has $(k,\beta_2)$-Poissonian box correlations, then it also has $(k,\beta_1)$-Poissonian box correlations.

Let $\alpha\neq 0$ and let $\theta>0$ satisfy $\theta\neq\tfrac12$ and $\theta\notin\mathbb Z$. For each $N$, let $y_1^{(N)}<\cdots<y_N^{(N)}$ be an ordering of the points…

Let $mu$ be the absolutely continuous limit measure of the numbers $\alpha H(n)\mod 1$, where $\alpha$ is the real root of $x^3+x=1$ and $H$ is Hofstadter's $H$ sequence. Regularity…

Maximum conjecture. If coprime bases $\beta_1$ and $\beta_2$ satisfy $\Upsilon(\beta_1)=u$ and $\Upsilon(\beta_2)=v$, then

Uniform-distribution conjecture. For every base $\beta$ with $\omega(\beta)=4$, there exists a $K$ such that, for all $k\geq K$, the digits in the sequences…

Greedy star-discrepancy conjecture. The sequence $\mathcal{S}^*$ should be a low-discrepancy sequence, meaning that

Greedy $L_2$ discrepancy conjecture. The sequence $\mathcal{S}^*$ should satisfy

Let $L$ be a number field and let $\alpha\in L$ be irrational. Let $\mathfrak{a}$ be an unramified ideal of degree one that is prime to the denominator of the principal ideal…

Let $Y$ be uniformly distributed on $[0,1)$, and for $N\in\mathbb{N}$ and $0<L\leq N$ define the random variable … Here $\operatorname{Po}(L)$ denotes a Poisson-distributed random…

Let $\mu$ be a non-atomic self-similar measure with support contained in $[1,\infty)$. A sequence $(y_n)_{n=1}^{\infty}$ is uniformly distributed modulo one if, for every interval…

Let $\alpha$ be an irrational number and $d$ be a real number. Consider the sequence $\{2^n d+n\alpha\mod 1\}_{n\geq 1}$ in $[0,1]$. Closure-interval conjecture. The topological cl…

Almost-everywhere obstruction conjecture. If for almost all $\alpha$ the pair correlations of $((\lbrace a_n\alpha\rbrace)_{n\in\mathbb N})$ are not Poissonian, then the pair corre…

Let $\left(a_n\right)_{n\geq 1}$ be a strictly increasing sequence of positive integers. It is quasi-arithmetic of degree $d$ if there exist constants $C,K>0$ and a strictly increa…

Let $\left(a_n\right)_{n\geq 1}$ be a sequence of distinct integers, and let $A_N$ denote its first $N$ elements. The additive energy of a finite set $A$ of reals is … where the su…

Let $\varphi(z)=\alpha_q z^q+\alpha_{q-1}z^{q-1}+\cdots+\alpha_0$ be a polynomial of degree $q$, and let $P_N(\,\cdot\,;\varphi)$ denote the gap distribution of…

Let $b$ be the base, let $\nu_N$ denote the empirical distribution associated with the fractional parts of $(\log_b n)$ up to $N$, let $E_b$ be the relevant exponential distributio…