---

Construction (p. 5). Let $2\leq m\leq n^{1/3}$ and $s=\sqrt{n/m}$. Take

$$\mathcal P=\{(a,0):1\leq a\leq m\},\qquad \mathcal Q=\{(i,\sqrt j):1\leq i\leq s,\ s^2+1-i^2\leq j\leq s^2+ms-i^2\},$$

with $a,i,j$ integers. Then $|\mathcal P|=m$ and $|\mathcal Q|=s\cdot ms=n$; the count, like the ranges, takes $s$ to be an integer, which the paper leaves implicit.

Statement. The printed proposition reads: "For the sets defined in (1), we have $D(\mathcal P, \mathcal Q) = \Theta(\sqrt{mn})$." (p. 5).

Proof pointer. Every squared cross distance is the integer $(a-i)^2+j$. Using $m\leq s$, which is where $m\leq n^{1/3}$ enters, these integers lie in an interval of length below $3ms=3\sqrt{mn}$, which gives the upper bound. The point $(1,0)$ alone has $ms=\sqrt{mn}$ distinct distances to the points $(1,\sqrt j)$, which gives the lower bound. The paper's displayed equality gives the number of integers in the interval as $3ms$; the exact number is $3ms-2m$, so $3ms$ holds only as an upper bound, and the conclusion is unaffected. This correction is this page's observation, not the paper's.

Remark 7 (p. 5) notes that for $m>n^{1/3}$ the same sets span $\Theta(m^2)$ distances, and Corollary 8 (p. 6) lists upper bounds on $D(m,n)$ by range of $m$, the second and third of them from this construction.

Source. Surya Mathialagan, On Bipartite Distinct Distances in the Plane, Electronic Journal of Combinatorics 28(4) (2021), P4.33, DOI 10.37236/9687: Section 2, the construction (1) and Proposition 6 with its proof on p. 5. The paper credits the construction to Elekes, Circle grids and bipartite graphs of distances, Combinatorica 15 (1995), 167--174. The copy read is identified on the source card.

Read depth. Claims checked: the construction, statement and proof were read clause by clause on the published PDF and the two bounds re-derived. Nothing here is independently reviewed.

Bears on. Problem 652: the problem page cites this restatement of Elekes's construction in its Formulation. In it each point of $\mathcal P$ determines at most $3\sqrt{mn}$ distances to $\mathcal Q$, which with Theorem 14 shows the order $\sqrt{mn}$ cannot be improved in that range. This page draws no conclusion about the problem's constants $\alpha_k$.