8 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 $I_C(m,n)$ denote the maximum number of incidences between $m$ points and $n$ circles. The circle-incidence conjecture. For some positive constant $c$, … This is a well-known i…

Let $C_m$ denote the variety associated with critical circles through a generic configuration of $m$ points in the plane, and let $EDD(C_m)$ denote its Euclidean distance degree. D…

Circle lattice-point conjecture. There is a constant $C_\alpha$ such that, for every $N$, the number of these lattice points is at most $C_\alpha$.

Let $P$ be a set of $n$ points in the plane in general position, meaning that no three are collinear and no four are cocircular. A pair of points has the required circle depth when…