55 problems

Let $P(x)$ be a polynomial, and let $\bar{\mathbf{d}}$ denote upper density. For a positive integer $T$, consider the set of integers $n$ for which $p-1$ divides $P(n)$ for some pr…

Positive-density reciprocal conjecture. If $A$ contains $0$, is not periodic, and is uniformly distributed modulo every power of $2$, then $\overline{A}$ has positive density. The…

Let $U$ be an avoidable set of integers, and suppose that $U$ contains infinitely many even numbers. Write $LD(U)$ for its lower logarithmic density. Lower-density conjecture. … Th…

Sparsest-obstruction conjecture. If

Let $[k]^d$ be the $d$-dimensional grid, with $k\in\mathbb{N}_{\geq 2}$ and $d\in\mathbb{N}$, and let $D(k,[k]^d)$ denote the maximum density of a configuration avoiding the releva…

Let $D(k,\mathbb{Z}^2)$ and $D(k, (\mathbb{Z}/k\mathbb{Z})^2)$ denote the corresponding maximum avoiding densities on the integer grid and the discrete torus, respectively. Torus-d…

Let $D(k,\mathbb{Z}^2)$ denote the maximum density of a subset of the integer grid avoiding the relevant $k$-in-a-row configurations, and let $3\mathbb{N}$ denote the positive mult…

Chen's conjecture. If $r_1,\ldots,r_t$ are positive integers with $r_1^{-1}+\cdots+r_t^{-1}\geq 1$, then $\mathcal{R}_{\mathcal{A}}\cap(\mathcal{R}_{\mathcal{A}}+2)$ has positive l…

Measure-sequence characterization. The sequences of measures defining a density in this sense are precisely those satisfying

Let $A\subseteq\mathbb{N}$ have positive density, where its density is … whenever this limit exists. Moreira's conjecture. There exist $x,y\in\mathbb{N}\backslash\{1\}$ such that ……

For a nonnegative integer $n$, an MC-circle is a largest circle enclosing exactly $n$ lattice points in its interior, and $R_n$ is its radius; an integer is MC if it is maximally c…

Let $f(k)$ and $g(k)$ be the integer-valued sequences defined in the paper's first and second special classes, respectively. Call a maximally circlable number strong if it is immed…

Let $f(x)=x^2-1$, and for an integer $n\geqslant 2$ let $P(n)$ denote the set of prime numbers arising in the associated sequence defined in the paper. Density-zero conjecture. For…

Cloitre's density conjecture. The number of $1$'s appearing in the prefix $a_1a_2\cdots a_n$ is $2n/3+o(n)$.

Let ${\mathsf{all}}$ be the ambient parameter set, and let $X,X'\subseteq{\mathsf{all}}$ satisfy … Let $Y$ be arbitrary, and write ${\mathfrak{dd}}_{X,Y}$ for the associated densit…

For $r\in(0,1)$, let ${\mathfrak{dd}}_{\{r\},{\mathsf{all}}}$ denote the corresponding density cardinal; the same notation is used when ${\mathsf{all}}$ is replaced by $[0,1]$,…

Let $\mathbb{M}$ be the cyclic free groupoid, and let $N$ be a finitely generated proper subgroupoid of $\mathbb{M}$. Null-density conjecture. The subgroupoid $N$ has null density.…

Hildebrand's nonemptiness conjecture. If $\mathcal{S}$ has positive lower density, then for every natural number $k$,

Hildebrand's conjecture. If $\mathcal{S}$ has positive lower density, then for every natural number $k$,

Multiplicative bounded-product conjecture. There exist an infinite set $B\subset\mathbb{N}$ and $t\in\mathbb{N}$ such that

Ordered linear sumset conjecture. There exist an infinite increasing set $B=\{b_1<b_2<\cdots\}\subset\mathbb{N}$ and an integer $t\geq0$ such that

Density threshold conjecture. The following assertions hold:

Infinite nested sumsets conjecture. There is an infinite sequence of infinite sets $B_1,B_2,\ldots\subset\mathbb{N}$ such that

Erdős's conjecture. For every $k\in\mathbb{N}$, there exist an integer $t\geq0$ and an infinite set $B\subset\mathbb{N}$ such that

Leader–Letzter–Narayanan–Walters conjecture. If $S\subset\mathcal{F}$ is product-free and $d^{\ast}(S)=1/2$, then