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Statement. For finite planar sets $P,Q$ with $|P|=m$, $|Q|=n$, $n\geq2$ and $n^{1/3}\leq m\leq n$, there is an absolute constant $c_0>0$ such that

$$D(P,Q)=|\{|p-q|:p\in P,q\in Q\}| \geq c_0\frac{\sqrt{mn}}{\log n}.$$

The sets may overlap, but each is a set of distinct points. Thus the minimum $D(m,n)$ over all such sets satisfies the same inequality. In particular,

$$D(n,n)=\Omega(n/\log n).$$

Source. Surya Mathialagan, On Bipartite Distinct Distances in the Plane, Electronic Journal of Combinatorics 28(4) (2021), P4.33, DOI 10.37236/9687. In the published PDF, the statement is on p. 3, the energy argument on pp. 9--12, and its remaining geometry on pp. 13--23. Physical and printed page numbers agree.

Current verification. Verified at the stated scope, retained in the final review and finalization-delta review. An independent source-based reviewer, distinct from the compiler, checked the complete own-words proof of the stated range and balanced specialization against the published Mathialagan source identified above. The living record covers Propositions 19--21, 27--28, 36, 40, 42, Corollary 37, Lemmas 25--26, the line/circle specialization of Lemma 34, and the external incidence interfaces. The compiler supplied the overlap, rotation-sign, projective-regulus and both-color counting repairs; the reviewer independently checked those arguments, their dependencies and their applications. No unresolved local proof gap remains within this scope.

The external premises are published Guth--Katz Theorems 1.2 and 4.5 in Annals 181 (2015), printed pp. 156 and 176. Their exact statements, edition, locators and local uses were independently checked; their proofs are neither compiled nor reviewed here. Theorem 4, Problem 652, the unused generality of Lemma 34 and the original constructibility route are outside this record. The lattice upper construction has its separate verification record on Theorem 1. This verification gives no whole-paper, problem-status or literature-freshness conclusion.

A substantive change to the source version, statement, argument, external premise or relied-on dependency returns the affected scope and its applications to Needs review until independently checked again.

Dependencies. The local proof is supplied on the following result pages.

• Proposition 19 gives the corrected positive-energy inequality.
• Proposition 20 and Proposition 21 classify the motions and bound translation energy.
• Proposition 27 gives the consistent rotation lines, overlap count and incidence bijection.
• Lemma 25 bounds point and plane concentrations.
• Lemma 26 bounds reguli, using the elementary line/circle distance lemma, projective-regulus proof, affine intersection exceptions, and complete circle and line ruling descriptions. The horizontal-line interpretation uses Proposition 28.
• The external incidence inputs state the exact Guth--Katz premises and prove their normalization.

Proof. Write $D=D(P,Q)$. First, if $D\geq\sqrt{mn}$, the desired bound holds after selecting $c_0\leq\log2$. We may therefore assume

$$D<\sqrt{mn}. \tag{1}$$

Use the positive energy $E$ of Proposition 19. Proposition 20 assigns to each of its quadruples the unique proper motion sending $p_1$ to $q_2$ and $q_1$ to $p_2$. Its translation part has cardinality at most $m^2n$ by Proposition 21. Let $E^{\rm rot}$ be the remaining part.

For the two labeled line families of Proposition 27, let $L$ be the underlying set of distinct spatial lines and $T=|L|$. With $s=|P\cap Q|$,

$$T=2mn-s^2,\qquad mn\leq T\leq2mn.$$

The given parameter range with $n\geq2$ forces $m\geq2$, so $T\geq4$. Lemma 25 bounds both point richness and plane concentration by $2m$. From (1) and Lemma 26 every regulus contains at most $8\sqrt{mn}$ lines. These are fixed multiples of $\sqrt T$, as required for the two-rich normalization of Guth--Katz.

Let $M_r$ count points incident to at least $r$ distinct lines of $L$. Only finitely many have $r\geq2$, since any two distinct lines have at most one intersection. Lemma 25 gives $M_r=0$ for $r>2m$. At an exactly $r$-rich point, the number of ordered intersecting pairs of distinct cross-color lines is $ab-c\leq r^2$, in the notation of Proposition 27. The energy bijection therefore gives

$$|E^{\rm rot}| \leq\sum_{r=2}^{2m}r^2(M_r-M_{r+1}) =4M_2+\sum_{r=3}^{2m}(2r-1)M_r. \tag{2}$$

The equality is finite summation by parts, using $M_{2m+1}=0$; the coefficient of $M_r$ is $r^2-(r-1)^2=2r-1$ for $r\geq3$.

Published Guth--Katz Theorem 1.2, with the padding proved in the incidence interface, gives $M_2=O(T^{3/2})$. Published Theorem 4.5, applied with $B=2m$, gives for every integer $r\geq3$

$$M_r\leq C\left(\frac{T^{3/2}}{r^2} +\frac{2Tm}{r^3}+\frac{T}{r}\right).$$

Substitution in (2), and $2r-1\leq2r$, bounds the rotation energy by an absolute constant times

$$T^{3/2} +T^{3/2}\sum_{r=3}^{2m}\frac1r +Tm\sum_{r=3}^{2m}\frac1{r^2} +T\sum_{r=3}^{2m}1 =O\!\left(T^{3/2}\log(2m)+Tm\right). \tag{3}$$

Here the harmonic sum is at most $1+\log(2m)$, and $\sum_{r=3}^{\infty}r^{-2}$ is bounded, for instance by comparison with the integral of $x^{-2}$. There are at most $2m$ terms in the final sum. Since $m\leq n$ and $mn\leq T\leq2mn$,

$$Tm\leq2m^2n\leq2(mn)^{3/2},\qquad \log(2m)\leq\log(2n)\leq2\log n.$$

Thus (3) is $O((mn)^{3/2}\log n)$. Translation energy is also absorbed: $m^2n\leq(mn)^{3/2}$ and $\log n\geq\log2>0$. Therefore

$$|E|=O((mn)^{3/2}\log n).$$

Finally Proposition 19 gives $D\geq(mn-s)^2/|E|\geq m^2n^2/(4|E|)$, yielding the required $c_0\sqrt{mn}/\log n$ with a uniform positive constant. All steps hold throughout the displayed source range. Substituting $m=n$ gives the balanced assertion.

Application to Problem 661. The lower bound has $\log n$ outside the square root. It is compatible with the requested $o(n/\sqrt{\log n})$ upper construction and does not disprove that question. The known lattice construction gives only $O(n/\sqrt{\log n})$. No mathematical status change follows.

Bears on. Problem 661.