Self-similar dimension from affine-map entropy

Source and verification

This article explains family 148 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Self-similar measures and exact overlaps

Consider finitely many affine contractions of the line,

$${\phi }_i(x)=r_{i}x+t_i,0<|r_i|<1,$$

with positive probabilities $p_i$. The associated self-similar measure is the unique probability satisfying

$$\mu =\sum _i{p}_i({\phi }_i{)}_{\ast }\mu .$$

Equivalently, choose independent symbols with law $p$ and compose their maps indefinitely. Contraction makes the limit well defined.

The family is indexed: two symbols may represent the same affine map. An exact overlap means that two distinct words of the same length give identical composed maps, including both signed slope and translation.

To account for this loss of information, let $G_n$ be the random composed affine map after $n$ symbols and set

$$\begin{array}{rl}h& =\underset{n\to \infty }{\lim }\frac{H(G_n)}{n},\\ \chi & =-\sum _i{p}_i{\log }_2|r_i|.\end{array}$$

All entropies here use base two. Since a length-$(n+m)$ map is determined by two independent blocks,

$$H(G_{n+m})\le H(G_n)+H(G_m).$$

Subadditivity gives the limit and identifies it with ${\inf }_{n}H(G_n)/n$. The contraction rate $\chi$ is positive.

🏷️ The claimed dimension formula

Statement in the manuscript

For every such finite indexed system,

$${\dim }_H\mu =\min \left\{1,\frac{h}{\chi }\right\}.$$

Ratios may be signed and unequal. Exact overlaps and repeated generators are allowed, with no separation or arithmetic assumption. (OpenAI, 2026)

The manuscript uses lower Hausdorff dimension, the infimum of dimensions of Borel sets with positive measure. An external exact-dimensionality theorem supplies an almost-everywhere constant local dimension; for these measures it also equals the full-measure-set dimension convention.

If distinct same-length words always give distinct maps, then $H(G_n)=nH(p)$ and the familiar entropy-over-contraction expression is recovered. The assertion concerns Hausdorff dimension. Dimension one does not assert absolute continuity.

🏷️ Repeated symbols change the relevant entropy

Take the maps $x\mapsto x/3$ and $x\mapsto x/3+2/3$ with equal weights. Their Cantor measure has no overlap, entropy rate one, and $\chi ={\log }_{2}3$, giving dimension $\log 2/\log 3$.

Now use three equally likely symbols, two specifying $x/3$ and one specifying $x/3+2/3$. The actual map probabilities are $2/3$ and $1/3$, so

$$\begin{array}{rl}h& =H_2(1/3)\\ & =-\frac{1}{3}{\log }_2\frac{1}{3}-\frac{2}{3}{\log }_2\frac{2}{3}.\end{array}$$

Using the symbol entropy ${\log }_{2}3$ would overcount information. The measure only sees which map acts, not which duplicate symbol was chosen.

For a single contraction, $h=0$ and the invariant measure is a point mass, consistent with dimension zero. The same zero-rate case is already settled by the elementary upper bound.

🏷️ The entropy upper bound

Fix a block length $n$ and treat each distinct map in $G_n$ as a new symbol, with its aggregated probability. The resulting system has the same invariant measure. Its symbol entropy is $H(G_n)$ and mean contraction depth is $n\chi$.

For long words of $j$ blocks, the strong law says that almost every address eventually has probability at least $2^{-j(H(G_n)+\epsilon )}$ and contraction at most $2^{-j(n\chi -\epsilon )}$. There are at most $2^{j(H(G_n)+\epsilon )}$ such typical prefixes.

Their images cover a full-measure set by intervals at the corresponding contraction scale. For

$$s>\frac{H(G_n)+\epsilon }{n\chi -\epsilon },$$

the total $s$th power of interval diameters tends to zero. Letting $\epsilon \downarrow 0$ and then taking the infimum over $n$ yields ${\dim }_H\mu \le h/\chi$. The ambient line gives the additional upper bound one.

This covering argument explains the easy direction. The research theorem says that no further dimension loss can occur.

🏷️ Information hidden at fine scales

Distinct maps may be extremely close without being equal. Absence of exact overlaps does not provide a quantitative minimum separation. The source therefore cannot simply choose a grid fine enough to distinguish all maps at a uniform exponential scale.

For a finite law, it compares exact entropy with the entropy of its grid cell. Their difference is information hidden inside cells. A finite-law estimate detects this hidden information through fair two-point distributions at finer scales, independently of the smallest positive distance between support points. (OpenAI, 2026)

Next, the proof groups symbols into blocks and conditions on their counts. Counts fix the product of the signed contraction ratios. Once this slope is fixed, the translation identifies the whole affine map.

The entropy cost of count conditioning is small compared with long block length. For an alphabet of size $m$, the number of count vectors is at most $(n+1{)}^m$, so its entropy is at most $m{\log }_2(n+1)=o(n)$. This preserves the linear entropy-rate information even though individual addresses can collide.

🏷️ Disjoint entropy gains and the contradiction

Assume a deficit below both one and $h/\chi$. The manuscript obtains a positive amount of fair-pair information in a finite band of very fine scales. Convolving with the unobserved tail turns each retained pair into a positive entropy gain.

The band endpoint may be arbitrarily deep. Successive block lengths are chosen only after earlier endpoints are known, rather than imposing an unavailable uniform separation bound.

At each observation scale, the retained blocks are disjoint. Their entropy gains therefore telescope to a bounded total. Averaging across observation scales produces a fixed positive contribution from every selected band, contradicting that bounded budget as the number of bands grows.

The disjointness and parameter order are indispensable: repeatedly charging the same hidden information would not prove a contradiction.

🏷️ Key steps and proof outlines

Hidden entropy is converted to fair-pair capacity

For a finite law $\nu$ on $F\subset \mathbb{R}$, define $c_{\ell }(\nu )$ as the maximum total weight of fair two-point laws supported on pairs at distance between $2^{-\ell }$ and $2^{1-\ell }$, whose sum is dominated by $\nu$. Equivalently, nonnegative pair weights satisfy $\frac{1}{2}{\sum }_{e\ni x}{w}_e\le \nu (\{x\})$. The half factor makes the constraints exactly those of a mixture of fair pairs. For $p{\delta }_x+(1-p){\delta }_y$, the capacity at its one distance band is $2\min (p,1-p)$.

The finite-law lemma bounds exact entropy hidden inside cells of width $\rho$ by

$$k(H(\nu )-H_{\rho }^0(\nu ))\le \log |kF|+C_k\sum _{\ell :2^{-\ell }\le 2\rho }{c}_{\ell }(\nu ).$$

Here $kF$ is the $k$-fold sumset and $H_{\rho }^0$ the entropy of the fixed width-$\rho$ cell containing a sample. To see the mechanism, take independent $X_1,\dots ,X_k\sim \nu$, reveal their initial cells and then their sum. The latter revelation costs at most $\log |kF|$. Successively refining the cells accounts for the remaining uncertainty by the entropy chain rule.

The cuts are perturbed randomly at every refinement level, independently of the samples. A pair of nearby atoms is then separated with probability bounded by its distance divided by the current cell scale, so an atom close to a deterministic boundary is not repeatedly overcharged. The source couples conditional refinements and converts the resulting distance-band mass into feasible fair-pair weights. For a fixed finite support, refinements eventually separate all atoms; no bound on that terminal depth appears in the final inequality. This is why extremely close but unequal maps do not invalidate the estimate. (OpenAI, 2026)

Conditioning fixes slopes at a sublinear entropy cost

The product of signed contraction ratios depends only on the count of each symbol. For an alphabet of size $m$, a length-$n$ count vector has at most $(n+1{)}^m$ possibilities and costs at most $m{\log }_2(n+1)$ bits. Conditional on it, a composed map is identified by its translation. The map entropy rate therefore still supplies linear translation information after this conditioning.

Under a hypothetical dimension deficit, the source combines this information with the finite-law pair bound to find bands $[a_n,B_n]$ carrying expected pair capacity at least $cn$. The endpoint $B_n$ may be arbitrarily large. Later block lengths are selected only after earlier endpoints are known, so the argument does not insert an unproved exponential separation estimate between distinct translations.

Retained windows charge each entropy loss once

Let ${\sigma }_j$ be accumulated contraction depth after $j$ environmental blocks, and fix an observation depth $L$. The adjusted depth $x_j=L-\lceil {\sigma }_j\rceil$ decreases with $j$. A candidate window of length $n_i$ is retained when its contraction depth is at most $A^{\prime }{n}_i$ and its adjusted depth belongs to $[a_{n_i},B_{n_i}]$.

Choose the bands so $a_{n_i}-A^{\prime }{n}_i-1$ exceeds the upper endpoints of all earlier, coarser bands. A retained window from band $i$ then ends while its adjusted depth is still above every coarser band. Monotonicity forces any retained coarser window to start afterward. Within one band, starts are multiples of its window length. Thus all retained windows are disjoint for every fixed environment and target depth.

If $g_j$ is the source’s nonincreasing conditional entropy quantity, disjointness gives

$$\sum _{\text{retained }[u,v)}(g_u-g_v)\le 2\log M,$$

where $M$ is the fixed entropy resolution parameter of this argument. This deterministic inequality is what prevents the same information from paying for several pair bands.

Averaging observation depths removes the band lengths

Choose a large target range $1\le L\le T$. Each band has at least $T/(2D{n}_i)$ candidate starts, where $D$ bounds contraction depth per block. Summing over $L$ visits every pair scale in that band’s interval exactly once at each start. Since the expected pair mass is at least $c{n}_i$, the length $n_i$ cancels. Every band contributes a fixed multiple of $T$.

The conditional entropy-gain lemma applies because retention is measurable with respect to the same block information on which that lemma conditions. Summing gives $2T\log M\ge \delta cqT/(4D)$ for $q$ selected bands and a fixed gain constant $\delta >0$. All constants were fixed before $q$, so choosing arbitrarily many bands is impossible. Independence between different retained windows is not used; disjointness, the tower property, and identical environment laws suffice.

See Also

Bernoulli convolution measure type addresses singularity and absolute continuity, beyond the dimension conclusion. Distance-set dimension supplies another setting where dimensional information is weaker than positive Lebesgue measure.

🏷️ Manuscripts and verification scope

• The entropy-rate dimension formula for self-similar measures on the line — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The scope includes the entropy-rate dimension formula with signed unequal ratios and exact overlaps, and selected no-overlap consequences for homogeneous measures and attractors. Absolute continuity is outside these statements. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. The entropy-rate dimension formula for self-similar measures on the line.