Full fixed-similarity orbits have unbounded rarity despite collisions

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Fix contractive planar similarities $f_1,\ldots,f_b$, allowing reflections, and a point $p$. Put

$$A_d=\{f_{w_1}\circ\cdots\circ f_{w_d}(p): w_i\in\{1,\ldots,b\}\},\qquad N_d=|A_d|.$$

Theorem. If $N_d\to\infty$, then $R(A_d)\to\infty$. If the generators do not have a common fixed point, then

$$R(A_d)\ge N_d^{1/6-o(1)}.$$

Unlike the dense-word theorem, this allows arbitrarily severe exact collisions. It concerns the full image sets of a fixed finite family, not arbitrary sparse selections or depth-dependent similarities. It does not resolve either unrestricted question of Problem 132.

1. Positive entropy and convex sumsets

Let $G_d$ be the distinct affine maps represented by words of length $d$. Write their generating linear parts as $r_iR_{\theta_i}J^{\epsilon_i}$, where $J$ is reflection in the real axis. A word's scale is a product $\prod r_i^{c_i}$ with $0\le c_i\le d$, and its angle is $\sum a_i\theta_i$ with $|a_i|\le d$. There are therefore at most

$$P(d)=2(d+1)^b(2d+1)^b$$

possible linear parts. A map is determined by its linear part and its value at $p$, giving

$$N_d\le |G_d|\le P(d)N_d.$$

Submultiplicativity $|G_{d+e}|\le|G_d||G_e|$ and Fekete's lemma show that $N_d=\exp(hd+o(d))$ for some $h\ge0$.

If two generators have distinct fixed points, choose a large closed ball containing $p$ and preserved by every generator. Sufficiently large equal powers of these two generators map it into disjoint neighborhoods of their fixed points. Words in these two blocks have pairwise distinct images of $p$: remove their common prefix and use injectivity and the disjoint images at the first differing block. Hence $N_{k\ell}\ge2^\ell$ for a fixed block length $k$. The limit defining $h$ gives $h>0$.

Translate $p$ to zero and split each word at $s=\lfloor d/2\rfloor$, $t=d-s$. Grouping prefixes by linear part gives

$$A_d=\bigcup_L(B_L+LA_t),\qquad B_L\subseteq A_s,$$

with at most $P(s)$ summands. The convex-sumset theorem checked in minkowski_sum_layers bounds every convexly independent subset of $P+Q$ by $C(|P|^{2/3}|Q|^{2/3}+|P|+|Q|)$. Thus every convexly independent subset of $A_d$ has size at most

$$H_d=CP(s)(N_s^{2/3}N_t^{2/3}+N_s+N_t) =N_d^{2/3+o(1)}.$$

The external input is Theorem 1 of Eisenbrand--Pach--Rothvoß--Sopher, Convexly independent subsets of the Minkowski sum of planar point sets. Its full statement imposes no separation or unique-representation condition; it was checked in the full paper in this attempt and in the earlier sumset argument.

Every convex vertex layer obeys this same bound. The rank--layer argument gives

$$R(A_d)\ge\left\lfloor\sqrt{N_d/(2H_d)}\right\rfloor =N_d^{1/6-o(1)}.$$

Rank existence follows exactly as in the selected-sumset proof. This also proves, for every fixed $\epsilon>0$,

$$B_d\subseteq A_d,\quad |B_d|\ge N_d^{2/3+\epsilon} \quad\Longrightarrow\quad R(B_d)\ge N_d^{\epsilon/2-o(1)}.$$

2. Common fixed points: normalization and counting

Move the common fixed point to zero. The case $p=0$ has bounded orbit, so assume $|p|=1$. Let $R$ be the largest contraction ratio, put $g_i=R^{-1}f_i$, and choose a ratio-one generator $g_0=U_0$. The sets

$$S_d=U_0^{-d}\{g_w(p):|w|=d\}$$

are similar to $A_d$, lie in the unit disk, and are nested: prepend $U_0$ to obtain $S_d\subseteq S_{d+1}$. If every generator has ratio one, $S_d$ lies on one circle; all its distance graphs have maximum degree two and at least $(N_d-1)/2$ distances occur. The conclusion follows.

Otherwise call a factor with ratio less than one subunit. We prove that there is a fixed integer $r\ge0$ such that

$$\#\{x\in S_d:x\text{ has a representation with at most }K \text{ subunit factors}\}=O_K((d+1)^r), \tag{1}$$ $$N_d\ge c(d+1)^{r+1}\quad(d\text{ sufficiently large}). \tag{2}$$

We use elementary finitely generated Abelian group counting. If an angular subgroup $\Gamma\subset S^1$ has torsion-free rank $r$, its bounded-exponent sets have $O((d+1)^r)$ elements. Positive products of at most $d$ generators have $\Omega((d+1)^r)$ elements: select $r$ generators with linearly independent free-coordinate vectors in $\Gamma\cong\mathbb Z^r\oplus F$.

There are three cases, with angles understood modulo $2\pi$.

A ratio-one reflection is available. Choose it as $U_0=J$. Let $\Gamma$ be generated by the angles of all ratio-one generators. A word with at most $K$ subunit factors has boundedly many possible subunit contributions; its remaining angular part is a bounded-exponent element of $\Gamma$. This proves (1). Even ratio-one words generate rotations $R_{2\theta}$ from direct generators and $R_\theta=(R_\theta J)J$ from reversing ones. Their subgroup has the same rank as $\Gamma$. Padding with $J^2$, and using a single $J$ for odd lengths, gives $\Omega(d^r)$ images at each sufficiently large exact length. For $0\le k\le d/2$, insert $k$ copies of one fixed subunit generator. These give distinct radii and $\Omega(d^r)$ points at each radius, proving (2).

All generators are direct. Write $U_0=R_\alpha$ and let $\Gamma$ be generated by $\theta_i-\alpha$ for the ratio-one generators. Commutativity and padding by $U_0$ give (1) and $\Omega(d^r)$ exact-length ratio-one images. Inserting $k$ copies of a subunit generator as above gives (2).

All ratio-one generators are direct, but a subunit reflection exists. Again write $U_0=R_\alpha$; take $\Gamma$ generated by $2\alpha$ and the ratio-one angles $\theta_i-\alpha$. If a word has $k\le K$ subunit factors, the signed number of its ratio-one letters has the parity of $d-k$. After subtracting $d\alpha$, its angular contribution is a bounded-exponent element of $\Gamma$, up to one of boundedly many offsets, including $-k\alpha$. This proves (1).

For (2), fix a subunit reflection $g=sR_\phi J$. For each odd $k\le d/2$, set $D=d-k$ and use words

$$U_{\rm left}\,g^k\,U_0^{D-\ell},$$

where $U_{\rm left}$ is any ratio-one word of length $\ell\le D$. Their varying normalized angles include all positive products of at most $D$ elements from $2\alpha$ and $\theta_i+\alpha$. These generate the same group $\Gamma$. Thus there are $\Omega(d^r)$ points at each of the $\Omega(d)$ distinct radii $s^k$, proving (2).

3. A finite surrounding set, or finitely many lines

In the reflection-reference case, consider the group generated by the angles of all generators. If it is finite, finitely many lines cover every $S_d$. If infinite, a direct generator's even powers or the block $(R_\theta J)J$ give a dense angular sequence among the nested sets $S_d$.

In the direct-only case use instead all angles $\theta_i-\alpha$. An infinite-order generator gives a dense sequence by its powers; a finite group again gives finitely many lines.

In the last case choose a subunit reflection angle $\phi_*$. The normalized directions lie in a bounded number of cosets of the group generated by

$$2\alpha,\quad \theta_i-\alpha\ (i\text{ direct}), \quad \phi_j-\phi_*\ (j\text{ reversing}).$$

If finite, this gives a uniform finite line cover. If infinite, a dense angular sequence comes from, respectively, words $g_*U_0^k$, powers of a direct generator, or powers of a pair of reflections. For the last case one may assume $2\alpha$ is torsion, since otherwise the first case already supplies the sequence; subtracting $2\alpha$ then preserves infinite order.

Finite line covers imply rarity growth by line_covers. In the remaining case, choose three directions of the dense sequence not contained in a closed semicircle. Their radii are positive, so their convex hull contains zero in its interior. Nesting gives a fixed $D$ and $\epsilon>0$ with

$$B(0,\epsilon)\subseteq\operatorname{conv}(S_D). \tag{3}$$

4. Negligible fixed layers in the zero-entropy case

Choose a subunit generator $g$ and $L$ so that the contraction ratio $\rho$ of $V=g^L$ is smaller than $\epsilon$. Put

$$C_j=U_0^{-(D+jL)}V^jU_0^D S_D.$$

Then $C_j\subseteq S_{D+jL}$. Each $C_j$ is an orthogonal image of $\rho^j S_D$, so (3) implies

$$C_{j+1}\subset\operatorname{int}\operatorname{conv}(C_j), \qquad B(0,\epsilon\rho^j)\subseteq\operatorname{conv}(C_j).$$

For sufficiently large $d$, $C_j$ survives the first $j$ convex vertex-layer deletions from $S_d$, by induction using the nested convex hulls. Every vertex of layer $j+1$ consequently has radius at least $\epsilon\rho^j$.

Let $\kappa<1$ be the largest subunit ratio. A point of that radius can use at most $\lfloor\log(\epsilon\rho^j)/\log\kappa\rfloor$ subunit factors. By (1)--(2), the $(j+1)$-st layer therefore has $O_j(d^r)=O_j(N_d/d)=o(N_d)$ points. Every fixed number of outer layers is negligible. The rank--layer theorem now makes every fixed number of largest distances rare eventually, completing the proof.

Checks and remaining scope

The author checked the positive-entropy argument, including exact overlaps and the polynomial count of linear parts; the three angular normal forms; the distinct-radius lower bounds; the normalization of the nested sets; and the rotation-independent nesting of $C_j$. The proofs were supplied by a separate author and are author-recorded.

This excludes full fixed-similarity word sets as bounded-rarity families even when $N_d/b^d\to0$. The positive-entropy selected-set bound above does not cover arbitrary sparse selections, and there is no reduction from arbitrary planar sets or nonstationary recursive constructions to this theorem.