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Source. Problem 2, p. 2 (Section 2, "The structure of point sets with few distinct distances", pp. 2--4), of Adam Sheffer, Distinct Distances: Open Problems and Current Bounds, arXiv:1406.1949v3 (2 July 2018), the edition read for the source card.

Statement

Definition (p. 2). A set $\mathcal P$ of $n$ points in $\mathbb R^2$ with $D(\mathcal P)=O(n/\sqrt{\log n})$ is called near-optimal; here $D(\mathcal P)$ is the number of distinct distances determined by $\mathcal P$.

Problem 2 (p. 2). "Characterize the near-optimal point sets." (quoted)

The survey's surrounding record (pp. 2--3). Sections of the integer lattice are near-optimal (via the Landau--Ramanujan count of sums of two squares, the survey's Theorem 2.1, p. 3), as is every $n$-point subset of a $c\sqrt n\times c\sqrt n$ section of $\mathbb Z^2$ for an integer $c\ge1$, and so are the rectangular lattices $\mathcal L_r=\{(i,j\sqrt r): i,j\in\mathbb Z,\ 1\le i,j\le\sqrt n\}$ for integers $r>1$ and their images under transformations such as rotation and uniform scaling. Erdős asked whether every near-optimal set "has lattice structure" (p. 2, quoted), and the author suggests (p. 3) that the near-optimal sets may be exactly those obtainable from the sets $\mathcal L_r$. The survey says hardly anything is known; the weaker question whether every near-optimal set has $\Omega(n^\varepsilon)$ points on a line is its Problem 3 (p. 3).

The introduction (p. 2) names characterizing the planar point sets that span few distinct distances (Section 2) as one of the two problems the author considers most challenging, the other being Problem 10.

Read depth

Claims checked on the print. The cited results are reported as the survey states them and were not checked against their sources here.

Bears on

No Erdős problem in this corpus is recorded here as bearing on Problem 2.