Rarity grows in dense fixed-similarity word constructions

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Fix $b\ge2$ contractive planar similarities

$$f_i(x)=r_iU_ix+t_i,\qquad 0<r_i<1,$$

allowing reflections, and fix a seed $p$. Write $f_w=f_{w_1}\circ\cdots\circ f_{w_d}$ for a word $w$ of length $d$. Suppose

$$A_d\subseteq\{f_w(p):|w|=d\},\qquad |A_d|\ge c b^d$$

for a fixed $c>0$. Then $R(A_d)\to\infty$.

Arbitrary selection and collisions are allowed subject to the displayed positive-density condition. The maps are fixed, not depth-dependent. This is a restricted construction theorem, not a solution of Problem 132.

The later full-orbit theorem removes the density assumption when $A_d$ is the entire set of word images. The theorem below remains useful for arbitrary selected subsets at positive formal density.

A measure lemma

Let $K$ be the compact attractor and $\mu$ the self-similar probability measure with uniform symbol weights. First suppose $K$ is noncollinear. Then every affine line and every boundary of a compact convex set has $\mu$-measure zero.

The measure is nonatomic. Otherwise its largest atomic mass $a>0$ is attained, and the set $M$ of atoms of mass $a$ is finite and nonempty. For $x\in M$, stationarity gives

$$a=\frac1b\sum_i\mu(\{f_i^{-1}(x)\}).$$

Every summand is at most $a$, so every preimage belongs to $M$. Thus each $f_i$ permutes $M$. Strict contraction of diameter makes $M$ a singleton fixed by all the maps. Their attractor would be that singleton, contrary to the assumption.

Let $\pi:\{1,\ldots,b\}^{\mathbb N}\to K$ be the coding map with Bernoulli measure. Suppose a convex boundary $\Gamma$ has positive $\mu$-measure. A planar convex boundary has at most countably many points without a unique tangent. Choose an address $\omega$ such that $x=\pi(\omega)\in\Gamma$ has a tangent line $L_x$, and the conditional-probability martingale for the event $\pi(\omega)\in\Gamma$ converges to one. For the prefix $w$ of length $d$, this says

$$\mu(f_w^{-1}\Gamma)\longrightarrow1. \tag{1}$$

Let $\rho_w$ be the contraction ratio of $f_w$ and $D=\operatorname{diam}K$. Since $x\in f_w(K)$, $f_w(K)\subseteq B(x,D\rho_w)$. Differentiability at $x$ therefore gives $\epsilon_d\to0$ such that

$$f_w^{-1}(\Gamma)\cap K \subseteq\{y:\operatorname{dist}(y,L_d)\le D\epsilon_d\}, \qquad L_d=f_w^{-1}(L_x). \tag{2}$$

Each $L_d$ meets $K$ at the coded tail point. A subsequence of these lines converges to a line $L$. By (1)--(2), every fixed neighborhood of $L$ has full $\mu$-measure, hence $\mu(L)=1$. Full support of $\mu$ implies $K\subseteq L$, a contradiction. When $\Gamma$ is an affine line the same argument uses zero strip width. Degenerate compact convex sets are contained in lines. This proves the lemma.

Dense distinct images have negligible convex layers

Define the word and distinct-point measures

$$\eta_d=b^{-d}\sum_{|w|=d}\delta_{f_w(p)},\qquad \nu_d=|A_d|^{-1}\sum_{a\in A_d}\delta_a.$$

Uniform contraction gives $\eta_d\Rightarrow\mu$. Every selected point has at least one word representative, so

$$\nu_d\le\frac{b^d}{|A_d|}\eta_d\le c^{-1}\eta_d.$$

Thus every weak subsequential limit $\nu$ satisfies $\nu\le c^{-1}\mu$.

Here is the needed convex-layer passage. Suppose finite sets $S_j$ lie in a fixed compact region, their sizes tend to infinity, and every weak limit of their uniform measures is dominated by a fixed multiple of $\mu$. If a positive proportion of their points lay on their convex-hull boundaries, pass to a subsequence with empirical limit $\nu$ and Hausdorff convex-hull limit $C$. The measure $\nu$ is supported on $C$. The measure lemma shows that $C$ has interior and $\nu(\partial C)=0$. Boundaries of convex bodies converging in Hausdorff distance to a body with interior converge to its boundary. All those boundary points therefore eventually lie in any prescribed closed $\delta$-neighborhood of $\partial C$. The portmanteau inequality and then $\delta\downarrow0$ contradict the positive boundary proportion.

Apply this observation to $A_d$. Its first vertex layer has $o(|A_d|)$ points. Removing that many points changes the normalized empirical measure by $o(1)$ in total variation. Repeating the observation proves

$$|B_k(A_d)|=o(|A_d|)\quad\text{for every fixed }k,$$

where $B_k$ is the union of the first $k$ convex vertex layers. No assertion that the post-deletion hull limit equals $\operatorname{conv}K$ is needed.

The rank--layer bound now gives

$$m_{A_d}(r_k)\le2k|B_k(A_d)|=o(|A_d|)$$

for the $k$-th largest distinct distance. Each fixed rank exists eventually by the two-anchor bound $n\le2q^2+2$. Hence any prescribed number of largest distances are eventually rare.

Degenerate attractors

If $K$ has at least two points and lies on a line $L$, every $f_i$ preserves $L$. In coordinates with $L=\{y=0\}$ its normal coordinate is $y\mapsto\beta_i y$. At depth $d$ that coordinate depends only on the symbol counts. Thus the word images are covered by at most $\binom{d+b-1}{b-1}$ parallel lines. This is $o(\sqrt{|A_d|})$, so even the elementary direct line-cover bound gives rarity growth.

If $K=\{o\}$, all maps fix $o$. A word image's radius about $o$ depends only on symbol counts. Its angle is an integer combination of the finitely many map angles, with coefficients in $[-d,d]$, together with a reflection parity. The number of distinct images is therefore polynomial in $d$, contradicting $|A_d|\ge c b^d$.

A separate author proved the measure lemma and layer passage; the author checked the martingale localization, nonatomicity, dominance, degenerate cases, and use of the rank--layer bound. The argument is author-recorded. This measure argument alone does not cover arbitrary sets, nonstationary similarities, or selected image sets of vanishing formal density. The later full-orbit result treats all full image sets, but not arbitrary sparse selections.