119 problems

Let $B(R)$ be the disc of radius $R$, and let … be the number of lattice points in it. Hardy's conjecture. One has … This is the central conjecture of the Gauss circle problem. The…

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

Let $K\subseteq\mathbb{R}^d$ be a convex body centred at the origin, meaning that its centre of mass is the origin. Assume that the interior of $K$ contains no lattice point other…

Let $M=\mathbb T^d$ be the standard flat torus, let $e_\lambda$ be a toral eigenfunction with frequency radius $\lambda$, and let $\Sigma\subset\mathbb T^d$ be a real-analytic hype…

Let $n$ be an integer representable as a sum of two squares, and consider lattice points on the circle of radius $\sqrt n$. For an arc of length $n^{\delta}$, the source states tha…

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…

Let $D(n)$ be the maximum number of lattice points that can be selected from the $n\times n$ square grid so that no three selected points lie on a common line. An obvious upper bou…

Let $R$ be a large parameter and define the thin annulus of lattice points … Assume this annulus contains approximately $R^{1/2}$ lattice points. Thin-annulus lattice-point restric…

Gauss sphere problem conjecture. It is conjectured that

Let $N\in\mathbb{N}$ and consider the circle centered at the origin with radius $\sqrt{N}$. An arc of length $N^{\eta}$, with $\eta<1/2$, is a short arc on this circle. Short-arc l…

For $\lambda>0$, consider lattice points on the circle $|x|=\lambda$ and arcs on that circle of length $\lambda^\delta$. Cilleruelo–Córdoba conjecture. For every $\delta<1$, there…

Randomness conjecture for Ripley's statistic. For squarefree $nneq 7\bmod 8$,

Let $\lambda S^1$ be the circle of radius $\lambda$, and let $M(\delta)$ denote a bound independent of $\lambda$ for the number of lattice points in an arc. Cilleruelo and Cordoba…

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…

Bleher–Lebowitz conjecture. A particular case of the conjecture of Bleher and Lebowitz states that $S_{\Lambda}(t,\rho)$ has a Gaussian distribution.

Let $B$ be the inside of a generic ellipse, let $\rho\to0$ with $\rho t\to\infty$, and define … For an ellipse $x_1^2+\alpha^2x_2^2=1$ with $\alpha$ Diophantine, let…

Let $B$ be the region being dilated, let $t$ be the inner radius, and let $\rho=\rho(t)$ be the annulus width. The microscopic regime is characterized by $\rho t$ being constant, a…

Let the width of a family of elliptical annuli tend to zero while the area of the annuli tends to infinity, and let the number of lattice points in the annulus be normalized to hav…

Bleher–Lebowitz conjecture. The normalized remainder $S(t,\rho)/\sigma$ has a standard Gaussian distribution.

Let $\sigma\subset\mathbb{Q}^n$ be a not necessarily convex cone, let $L^*\subset\mathbb{Q}^n$ be the dual lattice, and let $V=\{v_1,\ldots,v_m\}\subset L^*$. Let $S_\sigma$ denote…

The bounded dual-cone generators conjecture. $G$ is a semigroup basis for $S_\sigma$.

Relative discrepancy conjecture. In dimensions $d\geq 5$, the relative error satisfies $\Delta_{\Omega}(t)=O(t^{-2})$ as $t\to\infty$.