Dissonance of self-conformal measures on the real line
Expansions of real numbers
Let \(p \ge 2\) be an integer and \([x]_p = 0.x_1x_2\cdots\) the base-\(p\) representation of \(x \in [0,1]\), that is,
\[x = \sum_{i=1}^\infty x_ip^{-i}.\]
For fractions \(\frac{k}{p^n}\) there are two possible expansions—we choose the one which ends with \(0\)’s. If \(x \in \mathbb{Q}\), then the sequence of digits is eventually periodic in every base: for each \(p\) there are natural numbers \(m,k\) and blocks of digits \(y_1\cdots y_m\), \(x_1\cdots x_k\) such that
\[[x]_p = 0.y_1\cdots y_mx_1\cdots x_kx_1\cdots x_k\cdots.\]
If \([x]_p = 0.x_1x_2\cdots\) and \(q=p^2\), then
\[x = \sum_{i=1}^\infty x_ip^{-i} = \sum_{i=1}^\infty (px_{2i-1}+x_{2i})p^{-2i}\]
and hence, \([x]_p = 0.y_1y_2\cdots\), where \(y_i = px_{2i-1}+x_{2i}\). In a similar way, if \(q=p^n\), we obtain \([x]_q\) from \([x]_p\) by grouping digits into blocks of length \(n\). If there exist \(b,m,n \in \mathbb{N}\) such that \(p=b^m\) and \(q=b^n\) or, equivalently, \(\log p/\log q \in \mathbb{Q}\), then we say that integers \(p\) and \(q\) are multiplicatively dependent and write \(p \sim q\). In this case, the expansions \([x]_p\) and \([x]_q\) are closely related via the expansion in base \(k\). If \(\log p/\log q \notin \mathbb{Q}\), then we say that \(p\) and \(q\) are multiplicatively independent and write \(p \not\sim q\).
Minkowski dimensions
For a bounded set \(A \subset \R^d\), the \(r\)-covering number
\[N_r(A) = \min\biggl\{ k \in \mathbb{N} : A \subset \bigcup_{i=1}^k B(x_i,r) \text{ for some } x_1,\ldots,x_k \in \R^d \biggr\}\]
is the least number of closed balls of radius \(r>0\) needed to cover \(A\). The upper and lower Minkowski dimensions of \(A\) are
\[\begin{align*}
\overline{\dim}_{\mathrm{M}}(A) &= \limsup_{r \downarrow 0} \frac{\log N_r(A)}{\log r^{-1}}, \\
\underline{\dim}_{\mathrm{M}}(A) &= \liminf_{r \downarrow 0} \frac{\log N_r(A)}{\log r^{-1}},
\end{align*}\]
respectively. If the limit above exists, then the common value is the Minkowski dimension of \(A\) and we denote it by \(\dim_{\mathrm{M}}(A)\). Recall that \(\dim_{\mathrm{H}}(A) \le \underline{\dim}_{\mathrm{M}}(A) = \underline{\dim}_{\mathrm{M}}(\overline{A}) \le \overline{\dim}_{\mathrm{M}}(A) = \overline{\dim}_{\mathrm{M}}(\overline{A})\), where \(\overline{A}\) is the closure and \(\dim_{\mathrm{H}}(A)\) is the Hausdorff dimension of a \(A\). The partition of the real line into \(p\)-adic intervals is
\[\mathcal{I}_p = \biggl\{ \biggl[ \frac{k}{p}, \frac{k+1}{p} \biggr) : k \in \Z \biggr\}\]
and the corresponding partition of \(\mathbb{R}^d\) into \(p\)-adic cubes is
\[\mathcal{D}_p = \mathcal{D}_p^d = \{ I_1 \times \cdots \times I_d : I_i \in \mathcal{I}_p \}.\]
The covering number of \(A \subset \mathbb{R}^d\) by \(p\)-adic cubes is
\[D_p(A) = \#\{ Q \in \mathcal{D}_p : X \cap Q \ne \emptyset \}.\]
Since every \(p^n\)-adic cube is contained in a closed ball of radius \(\sqrt{d}p^{-n}\), we have \(N_{\sqrt{d}p^{-n}}(A) \le D_{p^n}(A)\). On the other hand, every closed ball of radius at most \(p^{-n}\) can be covered by at most \(3^d\) many \(p^n\)-adic cubes and we see that \(D_{p^n}(A) \le 3^dN_{p^{-n}}(A)\). Therefore, it is straightforward to conclude that for any integer \(p \ge 2\) we have
\[\begin{align*}
\overline{\dim}_{\mathrm{M}}(A) &= \limsup_{n \to \infty} \frac{\log D_{p^n}(A)}{\log p^n}, \\
\underline{\dim}_{\mathrm{M}}(A) &= \liminf_{n \to \infty} \frac{\log D_{p^n}(A)}{\log p^n}.
\end{align*}\]
Lemma 1.
If \(A \subset \mathbb{R}^d\) and \(B \subset \mathbb{R}^e\), then
\[\begin{align*}
\overline{\dim}_{\mathrm{M}}(A \times B) &\le \overline{\dim}_{\mathrm{M}}(A) + \overline{\dim}_{\mathrm{M}}(B), \\
\underline{\dim}_{\mathrm{M}}(A \times B) &\ge \underline{\dim}_{\mathrm{M}}(A) + \underline{\dim}_{\mathrm{M}}(B).
\end{align*}\]
In particular, if \(\dim_{\mathrm{M}}(A)\) and \(\dim_{\mathrm{M}}(B)\) exist, then \(\dim_{\mathrm{M}}(A \times B)\) exists and
\[\dim_{\mathrm{M}}(A \times B) = \dim_{\mathrm{M}}(A) + \dim_{\mathrm{M}}(B).\]
Proof. Notice that if \(Q \in \mathcal{D}_{p^n}^d\) is the unique \(p^n\)-adic cube containing \(x \in \mathbb{R}^d\) and \(R \in \mathcal{D}_{p^n}^e\) is the unique cube containing \(y \in \mathbb{R}^e\), then \(Q \times R \in \mathcal{D}_{p^n}^{d+e}\) is the unique cube containing \((x,y) \in \mathbb{R}^{d+e}\). Therefore,
\[D_{p^n}(A \times B) = D_{p^n}(A) \cdot D_{p^n}(B)\]
and the claim follows. ■
Dynamical interpretation
Let \(p \ge 2\) be an integer and \(f_p \colon [0,1] \to [0,1]\), \(f_p(x)=px \bmod 1\), the fractional part of multiplication by \(p\). If a closed non-empty set \(X \subset [0,1]\) satisfies \(f_p(X) \subset X\), then we say that \(X\) is \(\times p\)-invariant. For example, the closure of the forward orbit
\[O_p(x) = \{f_p^n(x) : n \in \mathbb{N}\}\]
of \(x \in [0,1]\) under \(f_p\) is \(\times p\)-invariant. Here \(f_p^n\) means the \(n\) times repeated composition \(f_p \circ \cdots \circ f_p\). Furthermore, if \(X \subset [0,1]\) is \(\times p\)-invariant, then there exists a collection \(F \subset \bigcup_{n=1}^\infty \{0,\ldots,p-1\}^n\) of forbidden sub-blocks such that
\[X = \{ x \in [0,1] : w \in F \text{ is not a sub-block of } [x]_p \}.\]
Conversely, such a collection \(F\) of forbidden sub-blocks defines a \(\times p\)-invariant set \(X\) by the above formula. For example, the choices \(p=2\) and \(F=\{1\}\) define the standard \(\frac{1}{3}\)-Cantor set as the \(\times 3\)-invariant set. We will next go through three technical lemmas. The first one guarantees that the Minkowski dimension exists.
Lemma 2.
If \(X \subset [0,1]\) is \(\times p\)-invariant, then \(\dim_{\mathrm{M}}(X)\) exists.
Proof. Notice that \(w=w_1\cdots w_n \in \{0,\ldots,p-1\}^n\) is the initial \(n\) digits of \([f_p^k(x)]_p\) for some \(k\) if and only if
\[f_p^k(x) \in \biggl[ \frac{z}{p^n}, \frac{z+1}{p^n} \biggr) \in \mathcal{I}_{p^n},\]
where \(z = \sum_{i=1}^n w_ip^{i-1}\). Since such a point \(z\) is in one-to-one correspondence with \(w\) and \(f_p(X) \subset X\), this imples that
\(D_{p^n}(X) = \#\{w\) is the initial \(n\) digits of \([f_p^k(x)]_p\) for some \(x\) and \(k\}.\)
Notice that if the concatenation \(vw\) of \(v \in \{0,\ldots,p-1\}^m\) and \(w \in \{0,\ldots,p-1\}^n\) is the initial \(m+n\) digits of \([f_p^k(x)]_p\) for some \(x\) and \(k\), then trivially \(v\) is the initial \(m\) digits of \([f_p^k(x)]_p\) and \(w\) is the initial \(n\) digits of \([f_p^{k+m}(x)]_p\). Therefore,
\[D_{p^{m+n}}(X) \le D_{p^m}(X) \cdot D_{p^n}(X)\]
for all \(m,n\) and, by Fekete’s lemma, \(\dim_{\mathrm{M}}(X)\) exists. ■
The second lemma further strenghtens the intuition that expansions of rational numbers are not that interesting and that multiplicatively dependent expansions are closely related.
Lemma 3.
(1) If \(p \ge 2\) is an integer and \(x \in \mathbb{Q} \cap [0,1]\), then \(\dim_{\mathrm{M}}(\overline{O_p(x)}) = 0\).
(2) If \(p \sim q\) and \(x \in [0,1]\), then \(\dim_{\mathrm{M}}(\overline{O_p(x)}) = \dim_{\mathrm{M}}(\overline{O_q(x)})\).
Proof. (1) If \(x \in \mathbb{Q} \cap [0,1]\), then there are natural numbers \(m,k\) and blocks of digits \(y_1\cdots y_m\), \(x_1\cdots x_k\) such that \([x]_p = 0.y_1\cdots y_mx_1\cdots x_kx_1\cdots x_k\cdots.\) Notice that \(w \in \{0,\ldots,p-1\}^n\) is a sub-block of \([x]_p\) if and only if \(w\) is the initial \(n\) digits of \([f_p^k(x)]_p\) for some \(k\). The proof of Lemma 2 thus shows that \(D_{p^n}(\overline{O_p(x)}) = \#\{\)distinct sub-blocks of length \(n\) in \([x]_p\}\). Therefore, \(D_{p^n}(\overline{O_p(x)}) \le m+k\) for all \(n\) and hence, \(\dim_{\mathrm{M}}(\overline{O_p(x)}) = 0\) as claimed.
(2) If \(p \sim q\), then there are \(b,m,n \in \mathbb{N}\) such that \(p=b^m\) and \(q=b^n\). Since \(f_q = f_{b^n} = f_b^n\), we have
\[\begin{align*}
\overline{O_b(x)} &= \overline{\bigcup_{i=0}^{n-1} f_b^i(O_{b^n}(x))} = \bigcup_{i=0}^{n-1} f_b^i(\overline{O_{b^n}(x)}) \\
&= \bigcup_{i=0}^{n-1} \bigcup_{I \in \mathcal{I}_{b^i}} (b^i(\overline{O_{b^n}(x)} \cap I) \bmod 1),
\end{align*}\]
where \(sA = \{sx : x \in A\}\). The finite stability of the upper Minkowski dimension and Lemma 2 give \(\dim_{\mathrm{M}}(\overline{O_b(x)}) = \max\{ \dim_{\mathrm{M}}(b^i(\overline{O_{b^n}(x)} \cap I)) \} = \dim_{\mathrm{M}}(\overline{O_{b^n}(x)}) = \dim_{\mathrm{M}}(\overline{O_q(x)})\). The proof is now finished since we similarly obtain \(\dim_{\mathrm{M}}(\overline{O_b(x)}) = \dim_{\mathrm{M}}(\overline{O_p(x)})\). ■
Finally, the third lemma allows us to switch to the Hausdorff dimension.
Lemma 4.
If \(X \subset [0,1]\) is \(\times p\)-invariant, then \(\dim_{\mathrm{M}}(X) = \dim_{\mathrm{H}}(X)\).
Proof.
By Lemma 2, it suffices to show that \(\dim_{\mathrm{H}}(X) \ge \dim_{\mathrm{M}}(X)\). Recall that the Hausdorff measure is positive, \(\mathcal{H}^s(X)>0\), if and only if the Hausdorff content is positive, \(\mathcal{H}^s_\infty(X)>0\). Therefore, it suffices to show that if \(s < \dim_{\mathrm{M}}(X)\), then \(\sum_j\textrm{diam}(U_j)^s > c > 0\) for all open covers \(\{U_j\}_j\) of \(X\). Since \(X\) is compact, we may assume that all open covers we consider are finite. Notice that if \(U \subset [0,1]\) and \(n\) is such that \(p^{-n-1} \le \textrm{diam}(U) < p^{-n}\), then there are \(I^1,I^2 \in \mathcal{I}_{p^n}\) such that \(U \subset I^1 \cup I^2\). It follows that every open cover \(\{U_j\}_j\) of \(X\) can be replaced by a cover \(\{I_i\}_i\) having twice the cardinality and consisting only of elements in \(\bigcup_n\mathcal{I}_{p^n}\). Therefore, as
\[\sum_i \textrm{diam}(I_i)^s = \sum_i p^{-ns} = 2\sum_j p^{-ns} \le 2p^s \sum_j \textrm{diam}(U_j)^s,\]
it is enough to show that \(\sum_i\textrm{diam}(I_i)^s \ge 1\) for all finite covers \(\{I_i\}_i\) of \(X\) consisting only of elements in \(\bigcup_n\mathcal{I}_{p^n}\).
Write \(R = \{w\) is a finite sub-block of \([x]_p\) for some \(x\}\). Recall from the proof of Lemma 2 that
\(D_{p^n}(X) = \#\{w\) is the initial \(n\) digits of \([f_p^k(x)]_p\) for some \(x\) and \(k\}.\)
We thus see that
\[D_{p^n}(X) = \#\{w \in R\text{ has length }n\}.\]
Furthermore, if \(\{w_i\}_i\) is a finite collection of elements in \(R\) such that for every \(v \in R\) having length large enough there are \(i\) and \(u\) such that \(v=w_iu\), then every \([x]_p\) begins with some \(w_i\). In other words, \(X \subset \bigcup_i I_{w_i}\), where \(I_{w_i}\) is the unique \(p^{n_i}\)-adic interval corresponding to \(w_i\) as in the proof of Lemma 2 and \(n_i\) is the length of \(w_i\). Hence it suffices to prove that \(\sum_i p^{-sn_i} \ge 1\) for all such finite collections \(\{w_i\}_i\).
Suppose to the contrary that there exists a collection \(\{w_i\}_i\) as above but with \(\sum_i p^{-sn_i}<1\). It follows that
\[\sum_{(i_1,\ldots,i_n)} p^{-s(n_{i_1}+\cdots+n_{i_n})} = \prod_{k=1}^n \biggl( \sum_{i_k} p^{-sn_{i_k}} \biggr) < 1.\]
By the defining property of the collection, there exists a finite family \(\{u_j\}_j\) such that every \(v \in R\) having length large enough can be expressed as a concatenation \(v=w_{i_1} \cdots w_{i_n}u_j\) for some sequence \((i_1,\ldots,i_n)\) and \(j\). Thus
\[\sum_n D_{p^n}(X) p^{-sn} = \sum_n \sum_{v \in R \text{ has length }n} p^{-sn} < \infty.\]
But since \(s < \dim_{\mathrm{M}}(X)\), we have \(D_{p^n}(X) > p^{sn}\) for all large \(n\) and the series above must diverge. This contradiction finishes the proof. ■
Furstenberg’s conjectures
Furstenberg conjectured that low complexity of the expansion of a given irrational number in one base \(q\) implies correspondingly high complexity in every other base \(p \not\sim q\).
Conjecture (Furstenberg (1970)).
If \(p \not\sim q\) and \(x \in [0,1] \setminus \mathbb{Q}\), then
\[\dim_{\mathrm{H}}(\overline{O_p(x)}) + \dim_{\mathrm{H}}(\overline{O_q(x)}) \ge 1.\]
Notice that Lemma 3 justifies the assumptions of the conjecture. Suppose that \(X \subset [0,1]\) is \(\times p\)-invariant and \(Y \subset [0,1]\) is \(\times q\)-invariant such that \(p \not\sim q\) and \(\dim_{\mathrm{H}}(X) + \dim_{\mathrm{H}}(Y) < 1\). If \(x \in X\), then \(\overline{O_p(x)} \subset X\) and \(\dim_{\mathrm{H}}(\overline{O_p(x)}) \le \dim_{\mathrm{H}}(X)\), and similarly for \(x \in Y\) and \(\overline{O_q(x)}\). If the conjecture holds and \(x \in (X \cap Y) \setminus \mathbb{Q}\), then
\[1 \le \dim_{\mathrm{H}}(\overline{O_p(x)}) + \dim_{\mathrm{H}}(\overline{O_q(x)}) \le \dim_{\mathrm{H}}(X) + \dim_{\mathrm{H}}(Y) < 1,\]
which is a contradiction. Therefore, the conjecture implies that \(X \cap Y \subset \mathbb{Q}\) and hence also \(\dim_{\mathrm{H}}(X \cap Y) = 0\). As very little is know about the validity of the conjecture, we will look at more geometric versions of the claim. We define \(sA = \{sx : x \in A\}\) and \(A+t = A+\{t\}\), where \(A+B = \{x+y : x \in A \text{ and }y \in B\}\).
Theorem 5 (Shmerkin (2019) and Wu (2019)).
If \(X \subset [0,1]\) is \(\times p\)-invariant and \(Y \subset [0,1]\) is \(\times q\)-invariant such that \(p \not\sim q\), then for any \(s,t \in \mathbb{R}\) we have
\[\dim_{\mathrm{H}}((sX+t) \cap Y) \leq \max\{ 0, \dim_{\mathrm{H}}(X) + \dim_{\mathrm{H}}(Y) - 1 \}.\]
In particular, if \(\dim_{\mathrm{H}}(X) + \dim_{\mathrm{H}}(Y) \le 1\), then the above result implies \(\dim_{\mathrm{H}}((sX+t) \cap Y) = 0\). Intuitively, this means that multiplication by \(p\) and multiplication by \(q\) have very little to do with each other. A “dual” version of Theorem 1 is the following:
Theorem 6 (Hochman-Shmerkin (2012)).
If \(X \subset [0,1]\) is \(\times p\)-invariant and \(Y \subset [0,1]\) is \(\times q\)-invariant such that \(p \not\sim q\), then for any \(s \ne 0\) we have
\[\dim_{\mathrm{H}}(X+sY) = \min\{ 1, \dim_{\mathrm{H}}(X) + \dim_{\mathrm{H}}(Y) \}.\]
Both theorems give positive answers to conjectures of Furstenberg from 1970 and late 60’s. Hochman’s lecture notes from 2014 give a detailed account for the proof of Theorem 6 and Furstenberg’s proof for Theorem 5 under the extra assumption \(\dim_{\mathrm{H}}(X) + \dim_{\mathrm{H}}(Y) < \frac{1}{2}\). Let us examine in detail in which sense the results are “dual” to each other. Fix \(e \in \mathbb{R} \setminus \{0\}\) and write
\[\mathrm{proj}_e \colon \R^2 \to \R, \quad \mathrm{proj}_e(x) = e \cdot x.\]
It is easy to see that \(\mathrm{proj}_e\) corresponds to the orthogonal projection onto the line \(\ell=\textrm{span}(e)\) spanned by \(e\). If \(X, Y \subset \mathbb{R}^2\) and \(s \ne 0\), then
\[\mathrm{proj}_{(0,1)}((X \times Y) \cap \ell_{s,t}) = (sX+t) \cap Y,\]
where \(\ell_{s,t} = \{(x,y) : y=sx+t\}\). Since \(\mathrm{proj}_{(0,1)}\) is a bi-Lipschitz map between \(\ell_{s,t}\) and the \(y\)-axis, the bi-Lipschitz invariance of the Hausdorff dimension gives us
\[\dim_{\mathrm{H}}((X \times Y) \cap \ell_{s,t}) = \dim_{\mathrm{H}}((sX+t) \cap Y).\]
Furthermore, since \(\mathrm{proj}_{(1,s)}(X \times Y) = X+sY\), we also see that
\[\dim_{\mathrm{H}}(\mathrm{proj}_{(1,s)}(X \times Y)) = \dim_{\mathrm{H}}(X+sY).\]
Therefore, the theorems are “dual” since the other is about slices and the other about projections of the planar set \(X \times Y\). We will next see that this connection is stronger than one would expect at first glance.
Marstrand’s slicing and projection theorems
Write \(\sigma^1 = \frac{1}{2\pi}\mathcal{H}^1\vert_{S^1}\) for the spherical measure on \(S^1\), where \(\mathcal{H}^1\) is the \(1\)-dimensional Hausdorff measure. The following result is Marstrand’s slicing theorem.
Theorem 7 (Marstrand (1954)).
If \(A \subset \mathbb{R}^2\) is a Borel set and \(e \in S^1\), then
\[\dim_{\mathrm{H}}(A \cap (\mathrm{span}(e) + x)) \le \max\{ 0, \dim_{\mathrm{H}}(A)-1 \}\]
for \(\mathcal{H}^1\)-almost all \(x\) in the orthogonal complement of \(\mathrm{span}(e)\).
The next result is Marstrand’s projection theorem.
Theorem 8 (Marstrand (1954)).
If \(A \subset \mathbb{R}^2\) is a Borel set, then
\[\dim_{\mathrm{H}}(\mathrm{proj}_e(A)) = \min\{1, \dim_{\mathrm{H}}(A)\}\]
for \(\sigma^1\)-almost all \(e \in S^1\).
Marstrand’s theorems give us a strong dimension conservation principle: either \(\dim_{\mathrm{H}}(A)\) is conserved by almost every projection or the surplus \(\dim_{\mathrm{H}}(A) - 1\) is stored in the fibers of the orthogonal projection.
Lemma 9.
If \(X \subset [0,1]\) is \(\times p\)-invariant and \(Y \subset [0,1]\) is \(\times q\)-invariant, then
\[\dim_{\mathrm{H}}(X \times Y) = \dim_{\mathrm{H}}(X) + \dim_{\mathrm{H}}(Y).\]
Proof. Let \(s < \dim_{\mathrm{H}}(X)\) and \(t < \dim_{\mathrm{H}}(Y)\). By Frostman’s lemma, there are Borel measures \(\mu\) such that \(\mu(X)>0\) and \(\mu(B(x,r)) \le r^s\) for all \(x \in \mathbb{R}\) and \(r>0\) and \(\nu\) such that \(\nu(Y)>0\) and \(\nu(B(x,r)) \le r^t\) for all \(x \in \mathbb{R}\) and \(r>0\). Since now \(\mu \times \nu(X \times Y)>0\) and \(\mu \times \nu(B((x,y),r)) \le r^{s+t}\) for all \((x,y) \in \mathbb{R}^2\) and \(r>0\), another application of Frostman’s lemma shows that \(\mathcal{H}^{s+t}(X \times Y)>0\) and \(\dim_{\mathrm{H}}(X \times Y) \ge s+t\). Since the choices of \(s\) and \(t\) were arbitrary, we get
\[\dim_{\mathrm{H}}(X \times Y) \ge \dim_{\mathrm{H}}(X) + \dim_{\mathrm{H}}(Y).\]
It follows from Lemma 1, Lemma 4, and the inequality above that
\[\begin{align*}
\dim_{\mathrm{M}}(X \times Y) &= \dim_{\mathrm{M}}(X) + \dim_{\mathrm{M}}(Y) = \dim_{\mathrm{H}}(X) + \dim_{\mathrm{H}}(Y) \\
&\le \dim_{\mathrm{H}}(X \times Y) \le \dim_{\mathrm{M}}(X \times Y).
\end{align*}\]
The claim follows since there is equality throughout. ■
By Lemma 9 and the earlier observations on how intersections of sets are connected to slices of their product, Theorem 5 thus claims that on the product \(X \times Y\) Marstrand’s slicing theorem extends to all lines \(\ell\) going through the origin besides the coordinate axes.
Theorem 5 (Shmerkin (2019) and Wu (2019)).
If \(X \subset [0,1]\) is \(\times p\)-invariant and \(Y \subset [0,1]\) is \(\times q\)-invariant such that \(p \not\sim q\), then we have
\[\dim_{\mathrm{H}}((X \times Y) \cap \mathrm{span}(e)) \leq \max\{ 0, \dim_{\mathrm{H}}(X \times Y) - 1 \}.\]
for all \(e \in S^1 \setminus \{(\pm 1,0),(0,\pm 1)\}\).
Similarly, since sums of sets are connected to orthogonal projections, Theorem 6 claims that on the product \(X \times Y\) Marstrand’s projection theorem extends to all orthogonal projections besides the ones onto the coordinate axes.
Theorem 6 (Hochman-Shmerkin (2012)).
If \(X \subset [0,1]\) is \(\times p\)-invariant and \(Y \subset [0,1]\) is \(\times q\)-invariant such that \(p \not\sim q\), then we have
\[\dim_{\mathrm{H}}(\mathrm{proj}_e(X \times Y)) = \min\{ 1, \dim_{\mathrm{H}}(X \times Y) \}.\]
for all \(e \in S^1 \setminus \{(\pm 1,0),(0,\pm 1)\}\).
Recall that the convolution of measures \(\mu\) and \(\nu\) is \(\mu \ast \nu = S_*(\mu \times \nu)\), where \(S(x,y) = x+y\). To prove Theorem 6 the task is to show that for every \(\times p\)-ergodic measures \(\mu\) and \(\times q\)-ergodic measures \(\nu\) we have
\[\dim_{\mathrm{H}}(\mu \ast (T_s)_*\nu) = \min\{1, \dim_{\mathrm{H}}(\mu) + \dim_{\mathrm{H}}(\nu)\},\]
where \(T_s(x) = sx\) and \(\dim_{\mathrm{H}}(\mu) = \inf\{\dim_{\mathrm{H}}(A) : \mu(A)>0\}\), and then rely on the variational principle, that is, on the existence of dimension maximizing ergodic measure. Recently, Yu (2021) and Austin (2021) gave new dynamical proofs of Theorem 5 and Glasscock-Moreira-Richter (2023) found a combinatorial proof of Theorem 6.
It is possible to approximate the closed \(\times p\)-invariant set \(X\) by a self-similar set \(X'\): For every \(\varepsilon > 0\) there is \(k \in \mathbb{N}\) and a \(p^{-k}\)-self-similar set \(X' \supset X\) such that \(\dim_{\mathrm{H}}(X) \ge \dim_{\mathrm{H}}(X') - \varepsilon\). In fact, this is how Wu begins his proof for Theorem 5. Here the \(n^{-1}\)-self-similar set means the homogeneous self-similar set in \([0,1]\) associated to a collection of similarities obtained by partitioning \([0,1)\) into \(n\) many subintervals. Already this gives motivation to study corresponding Furstenberg’s conjectures on self-similar sets. The question is interesting since we face an additional complication coming from possible overlapping. Peres-Shmerkin (2009) studied self-similar sets satisfying the strong separation condition, Nazarov-Peres-Shmerkin (2012) considered homogeneous self-similar measures with the strong separation condition, Hochman-Shmerkin (2012) had results also for self-similar measures under the strong separation condition, and Bruce-Jin (2022+) studied ergodic measures on homogeneous self-similar sets satisfying the strong separation condition.
In self-conformal sets, the defining contractions are \(C^{1+\alpha}\) maps instead of similarities. Let \(X\) and \(Y\) be self-conformal sets associated to \(\{f_i\}_i\) and \(\{g_i\}_i\), respectively. If there are \(i,j\) such that \(\log {Df_i\vert}_{p}/\log {Dg_j\vert}_{q} \notin \mathbb{Q}\), where \(p\) and \(q\) are the fixed points of \(f_i\) and \(g_j\), respectively, then we say that \(X\) and \(Y\) are multiplicatively independent. Quasi-Bernoulli measures are generalizations of Bernoulli measures and they include, for example, all Gibbs measures for Hölder continuous potentials. It is worth emphasizing that the following result does not assume any separation.
Theorem 9 (Bárány-K-Pyörälä-Wu (2023+)).
If \(X\) and \(Y\) are multiplicatively independent self-conformal sets, then
\[\dim_{\mathrm{H}}(\mu \ast \nu) = \min\{1, \dim_{\mathrm{H}}(\mu) + \dim_{\mathrm{H}}(\nu)\}\]
for all quasi-Bernoulli measures \(\mu\) on \(X\) and \(\nu\) on \(Y\).