A quantitative circular concentration threshold

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Subsequent improvement: weighted_circle_energy replaces the sufficient exponent $31/32$ below by $7/8$ and improves the diameter-inactivity exponent $1/96$ to $1/24$. The older proof is retained; its conclusions remain valid but weaker.

The circle incidence argument admits an explicit sublinear threshold. This remains a restricted result, not a solution of the two unrestricted questions.

Theorem. There is an absolute constant $C$ such that, if a circle contains $m\ge C(q(A)+1)^{31/32}$ points of $A$, then

$$R(A)\ge m/4. \tag{1}$$

Take all the points on the circle, with angular set $B$, as the anchor. Put

$$Q=q(A)+1,\qquad H=2Q/m,\qquad K=|B-B|/m.$$

The chord formula gives $1\le K\le H$. The pinned spectra used in the circle note have size at least $m/2$. For points at different positive radii about the anchor center, their overlap is at most

$$C_0(K^{4/3}m^{2/3}+K^2).$$

If $m\ge C_1H^7$, apply the union lower bound to $L=\lceil2H\rceil\le3H$ distinct radii. Their spectra would have union at least $Lm/2-O(H^{10/3}m^{2/3}+H^4)>Q$, a contradiction for large enough absolute $C_1$. Thus there are $k=O(H)$ occupied positive radii.

For their angular sets $B_i$, the covering argument gives

$$B_i\subseteq X_i+(B-B),\qquad |X_i|\le H.$$

Set $U=B-B$, $V=2B-2B$, $W=4B-4B$, with sizes $u,v,w$. The checked Plünnecke--Ruzsa input yields

$$u=Km,\qquad v\le K^4m,\qquad w\le K^8m.$$

The angular differences for any circle-pair type are covered by at most $H^2$ translates of $V$. Compare the anchor chord spectrum with just one such translate for a different unordered radius pair. The cosine relation is nondegenerate, as proved in the circle note. Its overlap lemma, using $U-U=V$ and $V-V=W$, gives at most

$$C_2\left(\frac{(vw)^{2/3}}{(uv)^{1/3}}+\frac wu+1\right) \le C_3(K^{19/3}m^{2/3}+K^7)$$

common values. There are $O(H^2)$ other radius-pair types. Including the at most $k=O(H)$ values incident with the possible center, only

$$O(H^{31/3}m^{2/3}+H^{11}+H)$$

anchor chord values occur outside the anchor circle. Each remaining positive chord value has all its pairs on that circle and therefore has unordered multiplicity at most $m\le|A|$. Consequently

$$R(A)\ge\frac{m-1}{2} -C_4(H^{31/3}m^{2/3}+H^{11}+H). \tag{2}$$

The condition $m\ge C_5H^{31}$, for a sufficiently large absolute constant, implies the earlier size requirement and makes (2) at least $m/4$. Since $H=2Q/m$, this condition is equivalent up to an absolute constant to $m\ge C Q^{31/32}$, proving (1).

Diameter inactivity tradeoff

Let $t(A)$ be the number of points not incident with a diameter pair. There is an absolute $c>0$ such that

$$R(A)+t(A)+1\ge c |A|^{1/96}. \tag{3}$$

Write $n=|A|$, $R=R(A)$, and $s=R+t+1$. Suppose $s\le c n^{1/96}$. For sufficiently small absolute $c$ this implies $t<n/2$, $q(A)+1\le n/2+R+1\le n$, and the size hypothesis below. The $m_0=n-t$ diameter-active points lie on a constant-width boundary, and their palette satisfies

$$q\le n/2+R\le m_0/2+(t/2+R).$$

Apply the quantitative rank inverse with $L=t/2+R$. Its parameter satisfies $E+1=\lceil2t+4R+4\rceil+1\le6s$. It produces a circle containing at least

$$m\ge\frac{n}{155520s^3}$$

points. The rank lemma's condition $m_0\ge1000(E+1)^3$ holds under the assumed bound on $s$, after reducing $c$ and absorbing small orders. The displayed $m$ exceeds $C(q(A)+1)^{31/32}$ when $c$ is sufficiently small. Formula (1) then gives

$$R\ge\frac{n}{622080s^3}>s,$$

a contradiction. This proves (3), with small $n$ handled by a further decrease of $c$.

In particular, bounded rarity forces every circle to contain $O(n^{31/32})$ points and forces $t(A)=\Omega(n^{1/96})$. Rarity diverges when $t(A)=o(n^{1/96})$. An arbitrary set may have linearly many inactive points, so neither unrestricted question follows.

The author checked the exponents, the uniform cosine incidence input and its degeneracies, and the rank-inverse application; the argument is author-recorded. All external ingredients and their full primary passages are already recorded in the linked source notes.