Quantitative short incidences and moving triangle frames

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These are necessary conditions for hypothetical bounded-rarity sequences, not a proof of either assertion of Problem 132. Constants may depend on a fixed rarity bound $K$ or fixed distance rank. The later anchor-multiplicity estimate improves the short-incidence exponent to $55/56$ and gives a polynomial convex-cover bound. The earlier calculation is retained below for comparison; the patch radius uses the improved estimate. The new weighted circle theorem gives

$$M(A):=\max_{\gamma\text{ circle}}|A\cap\gamma|=O_K(n^{7/8}) \quad\text{when }R(A)\le K.$$

Two external estimates

Write $\nu_C(g)=|C\cap gC|$ for a direct rigid motion. The estimates used are

$$\#\{g:\nu_C(g)\ge k\}\ll |C|^3/k^2\quad(2\le k\le |C|), \tag{1}$$

and, for a finite $S\subset\operatorname{Aff}(\mathbb C)$,

$$E(S)\ll V^{1/2}|S|^{5/2}+H|S|^2. \tag{2}$$

Here $E(S)$ counts $g_1g_2^{-1}=g_3g_4^{-1}$, and $V,H$ are the largest intersections of $S$ with vertical and nonvertical complex lines in the coordinates $g(z)=az+b$. These are equation (1) and Theorem 4.1 of Mansfield--Passant, A Structural Theorem for Sets with Few Triangles, full primary paper. The author checked both full statements; (2) is their stated theorem of Petridis--Roche-Newton--Rudnev--Warren. The use of (1) always separates one-point overlaps, since the family of such motions can be infinite.

Strict convexity also gives

$$r_C(v)\le2\ (v\ne0),\qquad E_+(C)\le3|C|^2-2|C|,$$

as proved in motion_incidence.

Quantitative rich motions on a convex set

Let $C$ be strictly convex, $|C|=m$, with at most $h$ points on any circle. Let $S$ consist of $s=O(m)$ distinct direct motions, each having at least $k\ge2$ overlaps with $C$. Then

$$k^5s^2\ll m^6\log(2m)(\sqrt s+h). \tag{3}$$

For a fixed rotation coefficient $a$, the corresponding overlaps are representation counts in $C-aC$. Their squared sum is $E_+(C,aC)\le E_+(C)\le3m^2$, by Cauchy--Schwarz. Thus $V\le3m^2/k^2$. A nonvertical line is a coset $g(x)=y$. For each $p\in C\setminus\{x\}$, its images under the distinct motions in this coset are distinct points of a circle centered at $y$. Hence $H(k-1)\le mh$, giving $H\ll mh/k$. Estimate (2) now gives

$$E(S)\ll (m/k)s^2(\sqrt s+h).$$

For the reverse estimate, count pairs $(g,x)$ with $x,gx\in C$. There are at least $ks$ such pairs. Cauchy--Schwarz over the $m$ possible sources gives at least $k^2s^2/m$ pairs of these actions sharing a source. For fixed $g,g'$, their number is at most $\nu_C(g'g^{-1})$. A second Cauchy--Schwarz, grouped by this quotient, gives

$$(k^2s^2/m)^2 \le E(S)\sum_{u\in SS^{-1}}\nu_C(u)^2.$$

By (1) and dyadic summation, the contribution with $\nu_C(u)\ge2$ is $O(m^3\log(2m))$; the contribution with $\nu_C(u)=1$ is at most $s^2=O(m^2)$. Consequently $E(S)\gg k^4s^4/(m^5\log(2m))$, proving (3).

Quantitative short incidences

Suppose $q(A)=O(n)$, a length $r$ has fixed smallest-distance rank $\ell$, and $E$ unordered $r$-edges touch a strictly convex subset. Partition that subset into the bounded number of narrow chains used in short_incident_inverse. One chain supports $\gg E$ oriented anchors; call its size $m$. The fixed-rank degree bound gives $m\gg E$.

In the narrow-chain proof, choose bin overload threshold $\asymp n/m$, rather than suppressing this dependence in a constant. Each anchor retains $\gg m$ apices, and there are $O(n^2/m)$ retained oriented triangle types. The same two Cauchy--Schwarz counts then supply

$$k\asymp m^3/n^2,\qquad s\asymp E m^2/n^2 \tag{4}$$

distinct $k$-rich motions, after taking a subset of them if needed. Here $s=O(m)$ since $E=O(m)$ by the fixed-rank degree bound. If either expression in (4) is too small to round to the required positive integers, then $E=O(n^{2/3})$, already below the bound below.

Substitution into (3) gives

$$E^2m^{13}\ll\log(2n) \left(n^{13}\sqrt E\,m+n^{14}h\right). \tag{5}$$

The first term, with $m\gg E$, gives $E\ll n^{26/27}(\log(2n))^{2/27}$. The second gives $E^{15}\ll n^{14}h\log(2n)$. Under $R(A)\le K$ the circle bound $h\ll_K n^{7/8}$ therefore yields

$$E\ll_{K,\ell} n^{119/120}(\log(2n))^{1/15}. \tag{6}$$

The constants are uniform over the convex subset and any fixed finite set of smallest-distance ranks.

Counting the multiplicity of each motion more carefully, rather than retaining just $O(m)$ motions, now gives the stronger bound

$$E\ll_{K,\ell}n^{55/56}(\log(2n))^{1/7}. \tag{6a}$$

The complete proof, including the rank dependence and one-point quotient overlaps, is in quantitative short incidences.

Growing patches when there is only one rare distance

Use the established one-rare reduction, with minimum separation normalized to one and $I=A\setminus(L_1\cup L_2)$. Let $b$ be the number of vertices of $I$ having fewer than six unit neighbors in $I$. Since $|I|<n/3$ and

$$e_1(I)\le3|I|-b/2,$$

richness of the minimum distance implies $b<2e_1(L_1\cup L_2,A)$. Formula (6a) bounds $b$ by its right-hand side.

Eventually every whole minimum-distance component lies in a triangular lattice, by the previously proved exact spectrum through length two. A radius-$L$ graph ball in such a lattice has $O(L^2)$ vertices. Thus only $O(bL^2)$ interior vertices lie within graph distance $L$ of a bad vertex. The others have complete radius-$L$ triangular patches. For sufficiently small fixed $c>0$, take

$$L=\left\lfloor c n^{1/112}(\log(2n))^{-1/14}\right\rfloor. \tag{7}$$

There are then $\gg n$ complete-patch centers. This improves the earlier fixed-radius statement to a polynomially growing radius. It still does not align the different patches.

For completeness, in any point-stabilizer coset of these canonically oriented patch frames there are $O(n/L^2)$ frames. Such a coset contains at most one frame of each orientation. Two radius-$\lfloor L/2\rfloor$ subpatches with different orientations cannot intersect: a shared point has a complete unit hexagon in both larger patches, whose directions must agree since the global minimum distance is one. These subpatches are therefore disjoint, each containing $\gg L^2$ points.

Many congruent triangles for every bounded rarity

Assume now only $R(A)\le K$ along an unbounded sequence. There are $\alpha_K,\beta_K>0$, a strictly convex $C\subset A$ with $|C|\ge\alpha_Kn$, a noncollinear rooted triangle $T_n$, and at least $\beta_Kn$ direct isometries $g$ such that

$$gT_n\subset A\setminus C,$$

with all roots $g(0)$ distinct.

Indeed the rank-layer bound gives a convex layer of size at least $\alpha_Kn$, where $\alpha_K=1/((K+1)(2K+1))$. Choose a rich length among the first $K+1$ smallest lengths. Its graph has maximum degree $D_K=6(K+1)$, and only $o(n)$ edges touch $C$, by (6), or already by the qualitative short-incidence theorem. Its induced graph on $A\setminus C$ therefore has at least $n-o(n)$ edges. At least $\alpha_Kn/(D_K-2)$ vertices have degree at least three, since otherwise the degree sum is less than $2(n-|C|)+(D_K-2)\alpha_Kn/(D_K-2)$.

There are $O_K(1)$ lengths at most $2r$. A radius-$2r$ disk is covered by $196$ squares of side $r/3$; each square has diameter less than $r$, hence at most $K$ internal lengths, and at most $2(K+1)^2$ points by the two-anchor bound. Thus the total number of pairs of length at most $2r$ is at most $196(K+1)^2n$. Since all but at most $K$ such lengths are rich, there are at most $J_K=K+196(K+1)^2$ occurring lengths in this interval.

Choose at each degree-three root two noncollinear $r$-neighbors. Pigeonhole their base length among these $J_K$ possibilities, then order the neighbors counterclockwise. This supplies $\beta_Kn$ directly congruent triangles with distinct roots. The triangle $T_n$ may vary with $n$ and may approach degeneracy; no contrary uniformity claim is made.

Expansion of the inverse-image union

More generally, let $|C|\ge\alpha n$ be strictly convex and let $G$ contain at least $\beta n$ direct motions with distinct roots $g(0)\in A$. Put

$$U=\bigcup_{g\in G}g^{-1}C,\qquad N=|U|,\qquad M=M(A).$$

Then

$$N^2(\sqrt N+M)\gg_{\alpha,\beta}\frac{n^3}{\log(2n)}. \tag{8}$$

To prove this, put $m=|C|$, $p=|G|\le n$. A same-orientation fiber with $v$ roots $B$ contributes a rotated copy of $C-B$ to $U$. Convex difference multiplicities and Cauchy--Schwarz give

$$N\ge |C-B|\ge\frac{m^2v}{m+2v-2};$$

hence $V\ll_\alpha N/n$. A nonvertical line $b=\lambda a+\mu$ in motion coordinates puts its distinct roots on $|b-\mu|=|\lambda|$ if $\lambda\ne0$; if $\lambda=0$ it contains at most one root. Thus $H\le M$. Applying (2) gives

$$E(G)\ll_\alpha n^2(\sqrt N+M).$$

On the other hand, if $r(z)=|\{g:z\in g^{-1}C\}|$, then

$$I:=\sum_zr(z)^2 =\sum_u r_{GG^{-1}}(u)\nu_C(u) \ge\frac{m^2p^2}{N}.$$

Cauchy--Schwarz and (1), again including at most $p^2$ one-point quotient overlaps, give

$$E(G)\gg_{\alpha,\beta}\frac{n^5}{N^2\log(2n)}.$$

This proves (8). Under bounded rarity $M\ll_K n^{7/8}$, so

$$N\gg_K \min\left\{ \frac{n^{6/5}}{(\log(2n))^{2/5}}, \frac{n^{17/16}}{\sqrt{\log(2n)}} \right\} \gg_K\frac{n^{17/16}}{\sqrt{\log(2n)}}. \tag{9}$$

Apply this to the triangle frames above. Every $z\in U$ has all three distances to $T_n$ in $D(A)\cup\{0\}$. The checked uniform three-anchor theorem in translated_triangles gives only $N=O_K(n^{12/7})$, using the updated Solymosi--Zahl input there. The two exponents are compatible. No stronger many-anchor estimate sufficient to contradict (9) was proved.

Collinear anchors alone cannot provide one: for fixed $L$, take $z_{x,k}=(x,\sqrt{R^2+k-x^2})$, $1\le x\le M_0$, $0\le k\le M_0$, and $R>(2L+2)M_0$. The squared distances to $(j,0)$, $0\le j\le L$, are $R^2+k-2jx+j^2$. They occupy only $O_L(M_0)$ values for $\Theta(M_0^2)$ distinct points.

There is a restricted many-anchor improvement: if the squared palette is covered by $r$ progressions of at most $Q$ terms, the uniform ellipse argument gives $N\ll_\varepsilon(q+1)r^2Q^\varepsilon$ and $M(A)\ll_\varepsilon r^2Q^\varepsilon$. Combined with (8), it forces $r\ge n^{1/10-o(1)}$ when $Q$ is polynomial in $n$. This does not control an arbitrary palette.

The author checked the displayed derivations, the external statements, the fixed-rank uniformity, the growing-patch argument, the circle improvement, and the two quantitative energy calculations; the arguments are author-recorded. These results do not close the unrestricted gap.