100 problems

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…

Erdős's conjecture. For every such $A$, there exist an infinite $B\subseteq A$ and an integer $t$ such that

Let $\mathscr{A}$ and $\mathscr{B}$ be two sets of positive integers. For a set $S$ of positive integers, write $S(x)=|S\cap[1,x]|$. Chen's Romanov-type conjecture. If there exists…

For a sequence of positive distances $(r_j)_{j\geq 1}$, let $\mathbf{m}_{\mathbb{R}^2}((r_j)_{j\geq 1})$ denote the supremum of the upper densities of sets in $\mathbb{R}^2$ avoidi…

Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be the two square-rim packings with an extra mass in one corner described in the example. The packing ${\mathcal{B}}$ is uniformly recurrent…

Two-saturated disk-packing density conjecture.

Let $d\geq 2$ and let $A\subseteq\mathbb{R}^d$ satisfy $\bar{d}(A)>0$, where $\bar{d}(A)$ denotes its upper limit density. Székely's conjecture. There is a $d_0$ such that every di…

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…

Szemerédi's conjecture. For every integer $k\geq 3$,

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…

Romani's conjecture. The asymptotic density of this set is approximately $0.434$. This is a quantitative question about the exceptional structure of the Romanoff sumset; the source…

Sparsest-obstruction conjecture. If

Let an instance $A=(a_i)_{i\in[k]}=(a_1,\ldots,a_k)$ consist of positive integer periods, and let its density be … A schedule assigns each task one day at a time while ensuring tha…

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 $k\in 3\mathbb{N}$, let $(\mathbb{Z}/k\mathbb{Z})^2$ be the discrete torus, and let $A\subseteq(\mathbb{Z}/k\mathbb{Z})^2$. The lines $L$ are the axis-parallel or diagonal line…

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…

Density conjecture. The set $4\mathcal{H}_14$ is dense in $(1,\infty)$.

Let $p$ be a prime, and consider pairs of positive integers $(m,n)$ in a reasonable planar region contained in the first quadrant. The relevant density is the probability that…

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 $\mathcal{P}$ denote the set of primes, and let $S=\{p+2^k:p\in\mathcal{P},\ k\in\mathbb{N}\}$ be the set of integers of the indicated form. Bombieri's conjecture. The set of o…