Quantitative short incidences and polynomial convex-cover bounds

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All multiplicities count unordered pairs. Write $n=|A|$, $q=q(A)$, $R=R(A)$, and $\lambda=\log(2n)$. Constants in $\ll$ below are absolute unless dependencies are indicated.

The improvement over moving frames is to count how often each motion is induced by an anchor pair: its multiplicity is at most $6j$ times its overlap with the convex chain, rather than merely the total number of anchors. This gives the following conclusions.

Short incidences. If $R(A)\le K$ and $r$ has fixed smallest-distance rank $j$, the number of $r$-edges touching any strictly convex subset is

$$O_{K,j}\bigl(n^{55/56}\lambda^{1/7}\bigr). \tag{1}$$

Convex covers. If $A$ is covered by $L\ge1$ strictly convex subsets, then for an absolute $c>0$,

$$R(A)+1\ge c\frac{n^{1/64}}{L^{7/8}\lambda^{1/8}}. \tag{2}$$

In particular a fixed convex cover gives polynomial rarity growth. More generally $L=o(n^{1/56}/\lambda^{1/7})$ forces $R(A)\to\infty$. Neither conclusion supplies such a cover for arbitrary planar sets.

Rich motions without a restriction on their number

Let $C$ be strictly convex, $|C|=m$, and suppose every circle contains at most $h$ points of $C$. For a finite family $S$ of $s$ direct motions having overlap $\nu_C(g)=|C\cap gC|\ge t\ge2$, we have

$$t^5s^2\ll m^3\bigl(m^3\log(2m)+s^2\bigr)(\sqrt{s}+h). \tag{3}$$

Here is the extension of the calculation in moving frames. The affine-group energy bound and the vertical/nonvertical fiber bounds $V\le3m^2/t^2$, $H\ll mh/t$ give

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

On the other hand, two applications of Cauchy--Schwarz give

$$\frac{t^4s^4}{m^2} \le E(S)\sum_{u\in SS^{-1}}\nu_C(u)^2.$$

The rich-motion estimate $\#\{u:\nu_C(u)\ge v\}\ll m^3/v^2$ for $v\ge2$ bounds the sum by $O(m^3\log(2m)+s^2)$; the last term accounts for one-point overlaps. Combining proves (3). No upper bound on $s$ was used.

The two external inputs are equation (1) and Theorem 4.1 of Mansfield--Passant, A Structural Theorem for Sets with Few Triangles. Their full statements were rechecked for this application. The remaining fiber estimates and the energy calculation are proved in the linked note.

An explicit narrow-chain estimate

Assume $q(A)\le n$, and let $r$ have smallest-distance rank $j$ in $A$. Let $C\subset A$ be an ordered strictly convex chain of $m$ vertices whose consecutive edge directions lie in an interval of width $\pi/3$. Suppose $F$ is a collection of $E\le n$ distinct ordered anchors $(p,q_0)$ with $p\in C$, $q_0\in A$, and $|p-q_0|=r$. Then

$$E\ll j^{17/12}n^{11/12} +j^{19/15}n^{14/15}\lambda^{2/15} +j^{8/7}n^{6/7}(h\lambda)^{1/7}, \tag{4}$$

where $h$ is the largest circular subset of $C$.

The case $E=0$ is immediate. The short-rank degree bound gives $E\le6jm$. Use the narrow-chain bin argument, now keeping the dependence on $j$ explicit. A radius interval of length $r$ has at most $20j$ pinned endpoints on the chain. Taking the bin overload threshold $\lceil240jn/m\rceil+1$ leaves, for $m\ge8$, at least $m/2$ noncollinear apices per anchor and at most $1452jn^2/m$ oriented triangle types. For $m\le7$, (4) follows directly from $E\le42j$; assume otherwise.

Thus the number of ordered pairs of retained congruent triangles is at least

$$P_0=c\frac{E^2m^3}{jn^2},\qquad c=1/6000. \tag{5}$$

Every ordered pair of anchors determines one direct motion. Let $f(g)$ count the anchor pairs inducing $g$, so $\sum_g f(g)=E^2$. Their number of matching apex pairs is at most $\nu_C(g)$ each. Discard motions with overlap less than

$$k=\frac{P_0}{2E^2}=\frac{cm^3}{2jn^2};$$

they account for at most $P_0/2$ matches. If $k<2$, then $m^3\ll jn^2$ and $E\le6jm\ll j^{4/3}n^{2/3}$, absorbed by the first term of (4).

The additional observation is

$$f(g)\le6j\nu_C(g). \tag{6}$$

Indeed, a source anchor root must lie in $C\cap g^{-1}C$, and there are at most $6j$ choices of its distance-$r$ neighbor. Once the source anchor is fixed, its image is fixed. This argument also applies to the identity motion.

Group the remaining generated motions into dyadic overlap intervals $[t,2t)$, starting at $k$, and write $s_t$ for the number in a level. There are at most $E^2\le n^2$ generated motions altogether. By (6),

$$P_0/2\le24j\sum_t t^2s_t. \tag{7}$$

Apply (3) to each level. If $k^5$ is not larger than a sufficiently large absolute constant times $m^3n$, then $m^{12}\ll j^5n^{11}$ and $E\le6jm$ give the first term of (4). Otherwise, since $\sqrt{s_t}+h\le2n$, the $s_t^2$ term on the right of (3) can be absorbed into the left for every $t\ge k$. We obtain

$$t^5s_t^2\ll m^6\lambda(\sqrt{s_t}+h),$$

and hence

$$s_t\ll m^4\lambda^{2/3}t^{-10/3} +m^3(h\lambda)^{1/2}t^{-5/2}. \tag{8}$$

Substituting (8) into (7) gives convergent geometric sums, not an additional logarithmic loss:

$$P_0\ll jm^4\lambda^{2/3}k^{-4/3} +jm^3(h\lambda)^{1/2}k^{-1/2}.$$

Insert (5) and the value of $k$. At least one of

$$E^6m^9\ll j^{10}n^{14}\lambda^2, \qquad E^4m^3\ll j^5n^6h\lambda$$

holds. Using $m\ge E/(6j)$ yields respectively the second and third terms of (4). This proves the estimate, including its rounding and low-overlap cases; real thresholds can equivalently be rounded after changing absolute constants.

Consequences for bounded rarity and convex covers

The weighted-circle theorem implies, when $q(A)\le n$,

$$M(A):=\max_\gamma|A\cap\gamma| \ll n^{7/8}+R(A). \tag{9}$$

Indeed, above its absolute $C(q+1)^{7/8}$ threshold a circular subset of size $h$ forces $R(A)\ge h/4$.

Under $R(A)\le K$, pair counting gives $q(A)\le n$ for large $n$. Partition any convex subset into at most twelve narrow chains and orient incident edges from that subset. Apply (4) and (9) to each chain. If there are more than $n$ anchors on a chain, first take $n$ of them; (4) excludes this for large $n$. Thus every chain has $O_{K,j}(n^{55/56}\lambda^{1/7})$ anchors, proving (1).

For (2), cases $R\ge\min(n/4,n^{7/8})$ already satisfy the result, after reducing the absolute constant. If every distance is rare, the two-anchor bound $n\le2q^2+2$ gives the result as well. In the remaining cases pair counting gives $q(A)\le n$. The first rich distance has rank $j\le R+1$. Select exactly $n$ of its unordered edges. Assign each edge to one endpoint, one of the $L$ covering convex subsets containing that endpoint, and one of its at most twelve narrow chains. Some chain has $n/(12L)$ assigned ordered anchors. Formula (4), with (9) and $R\le n^{7/8}$, gives at least one of

$$n\ll L^{12}j^{17},\qquad n\ll L^{15}j^{19}\lambda^2,\qquad n\ll L^{56}j^{64}\lambda^8.$$

The last right side dominates the first two because $L,j,\lambda\ge1$. Replacing $j$ by $R+1$ proves (2). One- and two-point covering sets cause no difficulty; they enter the bounded-chain-size case of (4).

If $b$ points are uncovered, the same argument gives

$$R(A)+1\gg\min\left\{\frac{n}{b+1}, \frac{n^{1/64}}{L^{7/8}\lambda^{1/8}}\right\}. \tag{10}$$

Indeed, either $6jb>n/2$, giving the first lower bound, or at least $n/2$ selected edges have a covered endpoint, and the same chain argument applies. Consequently the growing-cover divergence criterion also permits $o(n)$ exceptional points.

Finally, the patch argument in moving frames uses a bad-vertex count bounded by the short incidences. Replacing its older estimate by (1) gives complete triangular patches of radius

$$\left\lfloor c n^{1/112}\lambda^{-1/14}\right\rfloor$$

at linearly many centers in any hypothetical unbounded $R=1$ sequence. This still does not align the different lattices or close the unrestricted palette obstruction.

The author checked the motion multiplicity, generalized energy estimate, absorption, geometric sums, and all displayed exponents; the argument is author-recorded. No unrestricted conclusion follows.