45 problems

For an integer $n$, count integer points on the circle $a^2+b^2=n$ whose second coordinate lies in a short interval. The short-arc lattice-point conjecture. For every $\alpha<1/2$,…

Bourgain's circle congruence conjecture. There exist $\delta>0$ and an integer $m>0$ such that, if

The diagonal-arc conjecture. The number of these lattice points in an arc of length $R^{1-\epsilon}$ around the diagonal is bounded uniformly in $R$. The source presents this as th…

The circle-arc conjecture. The number of these lattice points in any arc of length $R^{1-\epsilon}$ is bounded uniformly in $R$. The source states the equivalent short-interval for…

Let $A_{k,d,\lambda}$ denote the maximal number of lattice points in the intersection of $k$ unit-width, $\nu$-transverse bands on the sphere $\lambda S^{d-1}$, with…

Visibility Density Conjecture. If $F(x)\in\mathbb{Z}[x]$ has at least two distinct roots, then the set of lattice points in $\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}$ visible along…

Betke-Henk-Wills conjecture. For any $K\in\mathcal{K}^d_0$ and $\Lambda\in\mathcal{L}^d$,

Let $F=(F_1,\ldots,F_p):\mathbb{R}^n\to\mathbb{R}^p$ be a polynomial map of degree $d$, let $K\subseteq\mathbb{R}^n$ be a compact set satisfying (P1), and let $T>0$ and $\alpha$ be…

Let $S(n,d)$ denote the number of $(d+2)$-tuples of points in $[n]^d$ that lie in a $(d-1)$-dimensional sphere. Sphere-counting conjecture. For any integer $d\geqslant 2$, … This c…

Additive-energy conjecture. For $\mathcal{A}_{\lambda,\delta}$,

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…

Call a nonnegative integer $n$ maximally circlable if there exists a largest circle enclosing exactly $n$ lattice points in its interior. Fix the key triangle with vertices…

No-$(d+2)$-on-a-sphere conjecture. There is an admissible subset of $[n]^d$ with

Parabolic-taxicab ball volume conjecture. The measure of this ball is

Let $\mathcal N=\{n\in\mathbb N:n\not\equiv 0,4,7\pmod{8}\}$, let $\mathcal E(n)=\{\mathbf{x}\in\mathbb Z^3:|\mathbf{x}|^2=n\}$, let…

Strengthened vanishing correlation conjecture. These vectors converge in distribution, in the double limit $R\to\infty$ followed by $k\to\infty$, to

Let $K\in\mathcal K^n$ be a full-dimensional convex body and let $\Lambda\in\mathcal L^n$. Let $K_s$ be the symmetrization of $K$, and let…

Let $K\theta?$.

Lattice polytope face-count conjecture. The number of $t$-dense faces of $\operatorname{conv}(S)$, counting faces of every dimension, is

Unit-circle-rootedness conjecture. Every complex root of $p_{n,m,\mathbf{c}}(x)$ lies on the unit circle $|z|=1$. This conjecture concerns the generating polynomials arising from l…

Full-density criterion for $V(1,1)$. If

Let $V(a_n,a_{n-1},\dots,a_1)$ denote the set of lattice points associated with the coefficient vector $(a_n,a_{n-1},\dots,a_1)$, and let $\operatorname{dens}$ denote its density.…

Let $d\geq 3$ be a natural number, let $m$ be a natural number, let $(\mathfrak{a}_{\bm{n}})_{\bm{n}\in S_{d,m}}$ be complex numbers, let $\epsilon>0$, and let…

For $N\in\mathbb{N}$, let $\mathbb{S}^1_+(N)$ be the upper semicircle of radius $\sqrt N$, and let $S_N=\mathbb{S}^1_+(N)\cap\mathbb{Z}^2$. Cilleruelo–Granville conjecture. For eve…