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Source. M. Charalambides, A note on distinct distance subsets, J. Geom. 104 (2013), no. 3, 439--442, DOI 10.1007/s00022-013-0176-0; read in the arXiv preprint arXiv:1211.1776v1, whose labels and page numbers are used here. Conjecture 2.3 and the paragraph before it on p. 2. The journal version was not compared. The edition read is identified on the source card.

Statement

With $\delta(N)$ as in Proposition 2.1:

Conjecture 2.3 (p. 2). "Given $\epsilon>0$, there exists some constant $c_\epsilon>0$ such that $\delta(N)\ge c_\epsilon N^{1/2-\epsilon}$."

The paragraph before it (p. 2) gives the motivation: the Guth--Katz bound is optimal up to constants, so the exponent $1/3$ seems to be the best the Lefmann--Thiele method gives in the form used, while the argument is wasteful, and the author expects the order of $\delta(N)$ to be closer to the upper bound of Proposition 1.2. The paper offers no proof and no further evidence.

Coverage

Claims checked: the conjecture and the paragraph before it were read on the page image of p. 2. Nothing here is independently reviewed.

Bears on. #1208, for $d=2$: the conjecture, if true, would give $F_2(N)\ge c_\epsilon N^{1/2-\epsilon}$ for every $\epsilon>0$, which with Proposition 1.2 would fix the exponent of $F_2(N)$ at $1/2$. It is a conjecture and proves nothing about the problem.