Background and proof strategy

The dimension of a self-similar set is classical when its pieces are sufficiently separated. Moran’s construction (Moran 1946) and Hutchinson’s iterated-function-system framework (Hutchinson 1981) give the similarity-dimension formula under the open set condition. Hutchinson also established the invariant-measure construction used above (Hutchinson 1981, Theorem 4.4(1)). Removing separation leads to the exact-overlap problem: can dimension fall below its natural upper bound when distinct words never define the same map? The set version is associated with Simon (Simon 1996); see also the formulation and historical discussion in (Varjú 2026, sec. 1). The entropy-rate formulation addresses the additional loss of information when exact overlaps do occur.

Bernoulli convolutions provide an early and influential model for this question. These are the laws of \(\sum_{j\ge0}\varepsilon_j\lambda^j\), where \(0<\lambda<1\) and the \(\varepsilon_j\) are independent equiprobable signs. Erdős proved singularity for reciprocal Pisot parameters in \((1/2,1)\) (Erdős 1939). A Pisot number is a real algebraic integer greater than one whose other conjugates have modulus less than one. Solomyak proved absolute continuity with an \(L^2\) density for almost every parameter in \((1/2,1)\) (Solomyak 1995); Peres–Solomyak later gave a simpler proof (Peres and Solomyak 1996). Garsia introduced the discrete entropy rate in this setting and related it to singularity (Garsia 1963, Theorem 1.2). These results illustrate the sensitivity to overlaps and arithmetic; absolute continuity and Hausdorff dimension are distinct questions.

Hochman showed that \(\dim_H\mu<\min\{1,H(p)/\chi\}\) forces superexponential concentration of cylinder maps (Hochman 2014, Theorem 1.1), using inverse theorems for entropy growth under convolution. For unequal ratios his entropy formulation already keeps track of the full affine map, including its slope (Hochman 2014, Theorem 1.4). The entropy-rate formula under weak exponential separation follows from his work; the version allowing exact collisions is proved in Bárány–Verma (Bárány and Verma 2026, Theorem 3.5). For Bernoulli convolutions, Breuillard–Varjú developed approximation by algebraic parameters with controlled entropy (Breuillard and Varjú 2019). Building on these and earlier entropy methods, Varjú proved full dimension for every transcendental parameter in \((1/2,1)\) (Varjú 2019b, Theorem 3). Rapaport proved the no-exact-overlap formula for algebraic contraction ratios and arbitrary real translations, allowing signed unequal ratios (Rapaport 2022, Theorem 2).

Rapaport–Varjú extended entropy and approximation methods to homogeneous three-map systems (Rapaport and Varjú 2024). For equal weights, their results include an exceptional parameter set of Hausdorff dimension zero and the no-overlap formula when \(\lambda>2^{-2/3}\) (Rapaport and Varjú 2024, Corollary 1.4 and Theorem 1.8). Rapaport–Varjú proved the entropy-rate formula, including exact overlaps, for rational translations and a positive common ratio (Rapaport and Varjú 2024, Theorem A.1). Feng–Feng proved the no-exact-overlap formula for algebraic translations and a signed common ratio (Feng and Feng 2025, Theorem 1.2). The distinction in Theorem 1 is that the entropy rate of the full affine-map walk is sharp even when exact collisions occur, with no arithmetic or separation assumption.

Absence of exact overlaps does not itself supply a quantitative separation estimate. Baker (Baker 2021, Theorem 1.3) and, independently, Bárány–Käenmäki (Bárány and Käenmäki 2021, Theorem 2.1) constructed systems without exact overlaps whose distinct cylinders approach one another at arbitrarily prescribed superexponential rates. Thus the entropy carried by extremely close maps must be handled without bounding the smallest positive separation.

Our proof combines the entropy retained by this map walk with a finite-law estimate that detects information across arbitrarily fine scales. It uses translation averaging, developed by Wang (Wang 2011, sec. 4.1), and the entropy calculus and fair-pair arguments of Varjú (Varjú 2019a, sec. 2.2, Propositions 20–21, and Lemma 22). Uniform nonsaturation has an antecedent for Bernoulli convolutions in Breuillard–Varjú (Breuillard and Varjú 2019, Lemma 13). We prove the required entropy estimates here; the only external dimension theorem used in the proof is exact dimensionality.

The argument has three steps. First, for a finite real law \(\nu\), compare its exact entropy with the entropy of its grid cell at a fixed scale. The difference is information hidden inside those cells. A bound for the sum of several independent copies forces this information to be witnessed by fair (equal-weight) two-point laws at smaller scales. The estimate is independent of the smallest distance between support points.

Second, group the coding symbols into blocks and condition on their symbol counts. This is the block-type disintegration of Galicer–Saglietti–Shmerkin–Yavicoli (Galicer et al. 2016, sec. 6.4 and Lemma 6.6), also developed by Saglietti–Shmerkin–Solomyak (Saglietti et al. 2018, Lemma 6.2); Käenmäki–Orponen (Käenmäki and Orponen 2023, sec. 2.2) explicitly allow repeated indexed maps in this construction. Here the counts fix each block’s signed contraction. For a sequence of such blocks, its translation then identifies its complete composed map. Conditioning costs only the entropy of the counts, so the map-entropy rate gives a lower bound for the remaining translation entropy even when addresses collide. The same fixed contractions give a small sumset bound. A hypothetical dimension deficit therefore yields a linear amount of fair-pair mass across a finite band of very fine scales.

Third, each pair produces a positive entropy gain against the unobserved tail. The bands may end at arbitrarily large depths. We choose successive block lengths only after the preceding endpoints are known. At each observation scale their retained blocks are disjoint, so their gains telescope to a bounded total. Averaging over observation scales gives a fixed positive contribution from every band, a contradiction as the number of bands grows.

Section 2 proves the entropy calculus, nonsaturation, and two-point gain. Section 3 proves the finite-law estimate. Section 4 applies it to conditional map laws, and Section 5 arranges the disjoint gains and completes the proof. Section 6 derives the no-overlap measure and attractor formulas, followed by the homogeneous and three-map specializations. No quantitative bound on the fine-scale endpoints is needed.

The entropy-rate upper bound

We prove the upper bound before turning to the contradiction argument. Fix a positive integer \(n\). Use the finite set of distinct maps in the support of \(G_n\) as a new alphabet, giving each map \(g\) its exact probability \(q_g=\mathbb P(G_n=g)>0\). Grouping independent coding symbols into \(n\)-blocks shows that this new system has the same measure \(\mu\). Its one-symbol entropy is \(H(G_n)\) and its mean contraction depth is \(n\chi\), since coincident maps have the same absolute slope.

For \(0<\varepsilon<n\chi\), the strong law shows that almost every address in this block system eventually has, at every length \(j\), a prefix of probability at least \(2^{-j(H(G_n)+\varepsilon)}\) and absolute contraction at most \(2^{-j(n\chi-\varepsilon)}\). There are at most \(2^{j(H(G_n)+\varepsilon)}\) such prefixes. For each fixed eventual starting length \(J\), let \(E_J\) be the closed set of addresses satisfying both conditions for every \(j\ge J\). Its compact coding image is covered by intervals of diameter at most \(C2^{-j(n\chi-\varepsilon)}\), where \(C\) bounds the attractor diameter. The total \(a\)th power of these diameters tends to zero when \(a>(H(G_n)+\varepsilon)/(n\chi-\varepsilon)\). The countable union over eventual starting lengths is a full-measure Borel set with this dimension bound. The case of zero attractor diameter is immediate. Letting \(\varepsilon\downarrow0\), then taking the infimum over \(n\), gives \[ d\le\min\{1,h/\chi\}. \tag{3}\] In particular, the conclusion of Theorem 1 already holds when \(h=0\), including a single-map system. To prove equality in all other cases, it suffices to rule out \[ d<1,\qquad d\chi<h. \tag{4}\] These inequalities are the contradiction hypothesis in Sections 4 and 5.

Parameters and conditional coding laws

Write \(m=|\Lambda|\). The second inequality in (25) implies \(h>0\), hence \(m\ge2\). Choose \(\alpha>\chi\) such that \(d\alpha<h\). For \(d>0\) we may take \(\chi<\alpha<h/d\); for \(d=0\) we may take \(\alpha=\chi+1\). Choose an integer \(b\ge1\) sufficiently large that \[m\log(b+1)<b\bigl(h-d\alpha\bigr),\] and set \[ A=b\alpha, \qquad A'=\frac{b(\chi+\alpha)}2, \qquad h_b=bh-m\log(b+1). \tag{26}\] Then \[ b\chi<A'<A, \qquad \gamma:=h_b-dA>0. \tag{27}\] These parameters remain fixed throughout the rest of the proof.

Group the iid symbols with law \(p\) into words \(W_0,W_1,\ldots\in\Lambda^b\). For \(w\in\Lambda^b\) let \(\operatorname{type}(w)=(\kappa_i)_{i\in\Lambda}\), where \(\kappa_i\) counts the occurrences of \(i\) in \(w\). The set \(\mathcal T_b\) of types consists of the nonnegative integer vectors with \(\sum_i\kappa_i=b\), and has cardinality at most \((b+1)^m\). For \(\kappa\in\mathcal T_b\), define \[N_\kappa=\frac{b!}{\prod_i\kappa_i!}, \qquad q_\kappa=N_\kappa\prod_i p_i^{\kappa_i}, \qquad Q_\kappa =\mathop{\mathrm{Law}}\bigl(W_0\mid\operatorname{type}(W_0)=\kappa\bigr).\] Every word of type \(\kappa\) has probability \(\prod_i p_i^{\kappa_i}\), so \(Q_\kappa\) is uniform on its \(N_\kappa\) words. Equivalently, we can generate the original coding by first choosing independent types \(\omega_0,\omega_1,\ldots\) with distribution \(q\), then choosing the words independently with respective conditional laws \(Q_{\omega_0},Q_{\omega_1},\ldots\). This follows by multiplying the probabilities of any finite sequence of blocks. In particular it preserves the original, possibly unequal, symbol weights \(p_i\).

Write \(\omega=(\omega_h)_{h\ge0}\) for the type sequence and \(\theta\omega=(\omega_{h+1})_{h\ge0}\) for its shift. For \(j\ge0\), let \(\mathcal F_j\) be the sigma-algebra generated by \(\omega_0,\ldots,\omega_{j-1}\); \(\mathcal F_0\) is trivial. For a type \(\kappa\) and an environment \(\omega\), define \[ r(\kappa)=\prod_{i\in\Lambda}r_i^{\kappa_i}, \qquad R_j=R_j(\omega)=\prod_{h<j}r(\omega_h), \qquad \sigma_j=-\log|R_j|, \tag{28}\] with \(R_0=1\) and \(\sigma_0=0\). Thus \(R_j\) is \(\mathcal F_j\)-measurable, including its sign. Let \[D=\max_{\kappa\in\mathcal T_b}\bigl(-\log|r(\kappa)|\bigr).\] The increments of \(\sigma_j\) are iid bounded positive random variables of mean \(b\chi\), and \(0\le\sigma_j\le Dj\).

Given \(\omega\), use the independent conditional block draws above to define \[ \nu_{\omega,n} =\mathop{\mathrm{Law}}_{\omega}\!\left(\sum_{h=0}^{n-1} R_h\varphi_{W_h}(0)\right), \qquad \mu_\omega =\mathop{\mathrm{Law}}_{\omega}\!\left(\sum_{h=0}^{\infty} R_h\varphi_{W_h}(0)\right). \tag{29}\] Here \(\mathop{\mathrm{Law}}_\omega\) denotes the law under those conditional word draws. The infinite sum converges uniformly, since the block translations form a finite set and \(\max_i|r_i|<1\). The law \(\nu_{\omega,n}\) depends only on \(\mathcal F_n\), whereas \(\mu_\omega\) can depend on the entire environment. Averaging over the types gives \(\mu=\mathbb E\mu_\omega\). For any real \(u\ne0\) and any law \(\lambda\), write \(\mathsf D_u\lambda=\mathop{\mathrm{Law}}(uX)\) when \(X\) has law \(\lambda\); this notation retains the sign of \(u\).

For \(n\ge1\), let \(\Omega_n=(\omega_0,\ldots,\omega_{n-1})\) be the tuple of the first \(n\) types, and put \[ \rho_n=2^{-\lceil An\rceil}, \qquad E_n=\{\omega:\sigma_n\le A'n\}, \qquad a_n=\lceil An\rceil-1. \tag{30}\]

From the deficit to bands of pair mass

The preceding lemma produces entropy that is invisible at scale \(\rho_n\). To apply the finite-law estimate, we must also control the number of possible sums of several independent copies of a conditional prefix. Fixing the types makes that count uniform in the environment.

The bands in Proposition 9 can extend far beyond the scale \(2^{-An}\). Their finiteness is enough: the next step will select successive block lengths after the earlier upper endpoints \(B_n\) are known. No quantitative separation of distinct cylinder translations has been used.

A gain after averaging the unobserved future

An individual conditional tail need not satisfy the nonsaturation bound for \(\mu\). To use that bound, we condition only on the types through the end of the finite block segment producing a fair pair. The unobserved future types then average its tail law back to a scaled copy of \(\mu\).

Choosing disjoint block windows

Fix any positive integer \(q\). Choose sufficiently large integers \(n_1<\cdots<n_q\) successively so that Proposition 9 applies, \(a_{n_i}\ge1\), and \[ a_{n_i}-A'n_i-1>B_{n_h}\qquad(h<i). \tag{42}\] This is possible because the previously chosen \(B_{n_h}\) are finite and \[a_n-A'n-1=\lceil An\rceil-A'n-2 \ge(A-A')n-2\longrightarrow\infty.\] In particular, the choice uses no upper bound on how fast \(B_n\) grows. Call \([a_{n_i},B_{n_i}]\) band \(i\). Larger \(i\) means finer physical scales, since (42) gives \(a_{n_i}>B_{n_h}\) for \(h<i\).

Choose an even integer \(T\) with \[ \frac T2>\max_{1\le i\le q}B_{n_i}. \tag{43}\] For band \(i\), use the candidate starts \[ J_i=\left\{kn_i:k\in\mathbb Z,\ 0\le k\le \left\lfloor\frac{T}{2Dn_i}\right\rfloor\right\}. \tag{44}\] For each target \(L\in\{1,\ldots,T\}\), retain the window \([j,j+n_i)\), \(j\in J_i\), exactly on the event \[ \mathcal A_{i,j,L}= \left\{\sigma_{j+n_i}-\sigma_j\le A'n_i,\quad a_{n_i}\le L-\lceil\sigma_j\rceil\le B_{n_i}\right\}. \tag{45}\] This event belongs to \(\mathcal F_{j+n_i}\), the same sigma-algebra as the conditioning in Lemma 10.

The reason for the separation condition is simple. As a block start \(j\) moves forward, the adjusted target depth \(L-\lceil\sigma_j\rceil\) is nonincreasing. A retained window in a finer band cannot decrease this depth far enough to reach any coarser band. The next lemma makes this order precise for every environment.

Figure 1 shows the order of retained windows from two different bands in this proof.

Summing over target depths

The disjointness lemma bounds the total gain at one physical scale. Summing over targets recovers every pair scale in each band. Indeed, for a candidate \(j\in J_i\), \[0\le\sigma_j\le Dj\le T/2.\] Because \(T\) is even, this also gives \(0\le\lceil\sigma_j\rceil\le T/2\). Each integer \(\ell\in[a_{n_i},B_{n_i}]\) therefore occurs exactly once as \(L-\lceil\sigma_j\rceil\) for \(1\le L\le T\): take \(L=\ell+\lceil\sigma_j\rceil\). Its lower bound is \(1\), and its upper bound is at most \(B_{n_i}+T/2<T\).

Each band has at least \(T/(2Dn_i)\) candidate starts, and each contributes expected pair mass at least \(cn_i\) after summing the targets. The block length cancels, so every band supplies a fixed multiple of \(T\). Take expectations in (46). Since the retention indicator is \(\mathcal F_{j+n_i}\)-measurable, the tower property and Lemma 10 apply to each summand. Sum the result over \(L=1,\ldots,T\) and use the preceding exact correspondence to obtain \[\begin{align*} 2T\log M &\ge\frac\delta2\sum_{i=1}^q\sum_{j\in J_i} \mathbb E\left[ \mathbf1_{E_{n_i}}(\theta^j\omega) \sum_{\ell=a_{n_i}}^{B_{n_i}} c_\ell(\nu_{\theta^j\omega,n_i})\right] \\ &\ge\frac{\delta c}{2}\sum_{i=1}^q |J_i|n_i \ge\frac{\delta c qT}{4D}. \tag{48}\end{align*}\] The second inequality uses Proposition 9 and the identical distribution of the type environment under \(\theta^j\); independence between different candidate windows is not needed. The last inequality follows from the exact count \[|J_i|=\left\lfloor\frac{T}{2Dn_i}\right\rfloor+1 \ge\frac{T}{2Dn_i}.\] After canceling \(T\), (48) bounds the arbitrary integer \(q\) by \(8D\log M/(\delta c)\). All the constants in this bound were fixed before \(q\) was chosen, so this is impossible.