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Every scale and every center

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

The independent tests of Random routing and independent tests control one scale \(t\) at one stable center \(x\). This lesson controls all scales \(t\in[1,2]\) at once, chooses the parameters of the construction, and repairs the centers where the random set still misses by a small open set [OpenAI-ES, Section 5]. This proves Proposition 3.1 of Measure-universal sets and periodic hitting sets, and with it Theorem 2.2 there: no geometric progression is measure universal.

We keep the notation of Dyadic grids and windows on a tree and Random routing and independent tests. From the core courses we use continuity of measure from above (Fremlin, Measure Theory, Volume 1, 112C, in Measure and Integration) and the Bolzano–Weierstrass theorem (Lebl, Basic Analysis I, Theorem 2.3.8, in Real Analysis I). For an integer \(r\ge1\) put \[ \begin{aligned}D_{M,q}(r)&=4+2(M-1)r\\&\quad\times\bigl(1+2c\,q^{-2r}\bigr).\end{aligned}\tag{0.1} \]

1. All scales at a stable center

Lemma 1.1 (all scales; [OpenAI-ES, Lemma 5.1]). Let \(x\in\mathcal S\) be a stable center and \(\xi\) an atom of its exposure on which the route of \(x\) makes a default choice; let \(U\) be the first vertex where it does, \(h\) its height and \(r=r_h\). Then \[ \begin{aligned}&\mathbb P\Bigl(\exists t\in[1,2]\ \forall n\in\mathcal N:\\&\qquad x+tq^n\notin B\Bigm|\xi\Bigr)\\&\quad\le D_{M,q}(r)\,e^{-p(M-1)r/2}.\end{aligned}\tag{1.1} \]

Proof. On \(\xi\) the vertex \(U\) and the edges \(e_i=(U,i)\), \(i<M\), are fixed. The local test \(Q_i(z)\) of (3.1) of Random routing and independent tests reads the key of \(z\) on \(e_i\) and keys of \(z\) on edges below \(U_i\) and at leaves below \(U_i\); all of them have resolution at most \(b_{e_i}^*\). By Lemma 1.1(2) of Dyadic grids and windows on a tree, for every outcome \(Q_i(z)\) is a function of the single key \(J_{b_{e_i}^*}(z)\).

Recorded scales. For \(i<M\) and \(n\in W_{e_i}\), let \(\mathcal T_{i,n}\) be the set of \(t\in[1,2]\) with \(x+tq^n\in\Gamma_{b_{e_i}^*}\). As \(t\) runs over \([1,2]\), the point \(x+tq^n\) runs over an interval of length \(q^n\), which contains at most \(1+q^nN_{b_{e_i}^*}\) points of the lattice \(\Gamma_{b_{e_i}^*}\). By Lemma 1.1(1) and Lemma 2.1 of Dyadic grids and windows on a tree, with \(n\ge a_{e_i}\) and \(b_{e_i}^*\le a_{e_i}+2r-1\), \[ q^nN_{b_{e_i}^*}<2c\,q^{n-b_{e_i}^*}\le2c\,q^{1-2r}\le2c\,q^{-2r}. \] Let \(\mathcal T_0\) be the union of the sets \(\mathcal T_{i,n}\) with \(\{1,2\}\); there are \((M-1)r\) pairs \((i,n)\), so \(|\mathcal T_0|\le2+(M-1)r(1+2cq^{-2r})\). Let \(\mathcal T\) consist of \(\mathcal T_0\) and the midpoints of consecutive elements of \(\mathcal T_0\); then \(|\mathcal T|\le2|\mathcal T_0|\le D_{M,q}(r)\). The set \(\mathcal T\) depends only on \(x\), the windows, the grids and \(U\); it is fixed on \(\xi\).

Constancy between recorded scales. Let \(J\) be an open interval between consecutive elements of \(\mathcal T_0\), and \(t_1<t_2\) in \(J\). A point of \(\Gamma_{b_{e_i}^*}\) in \((x+t_1q^n,\,x+t_2q^n]\) would be \(x+t'q^n\) with \(t'\in(t_1,t_2]\subseteq J\), a recorded scale inside \(J\). So by Lemma 1.1(3) of Dyadic grids and windows on a tree, the key \(J_{b_{e_i}^*}(x+tq^n)\) is constant for \(t\in J\), and for every outcome the whole vector \(\bigl(Q_i(x+tq^n)\bigr)_{i<M,\ n\in W_{e_i}}\) is constant on \(J\).

Union bound. Suppose that for some \(t\in[1,2]\) no point \(x+tq^n\), \(n\in\mathcal N\), lies in \(B\). By Lemma 3.1 of Random routing and independent tests, all local tests fail at \(t\). If \(t\in\mathcal T_0\), put \(t'=t\); otherwise \(t\) lies in an open interval between consecutive elements of \(\mathcal T_0\), because \(1,2\in\mathcal T_0\), and we let \(t'\) be its midpoint. In both cases \(t'\in\mathcal T\) and all local tests fail at \(t'\). Each \(t'\in\mathcal T\) is a fixed number on \(\xi\), so Lemma 4.1 of Random routing and independent tests applies to it, and \[ \begin{aligned}&\mathbb P\Bigl(\exists t\in[1,2]\ \forall n\in\mathcal N:\\&\qquad x+tq^n\notin B\Bigm|\xi\Bigr)\\&\quad\le|\mathcal T|\Bigl(1-\frac p2\Bigr)^{(M-1)r}\\&\quad\le D_{M,q}(r)\,e^{-p(M-1)r/2},\end{aligned} \] using \(1-u\le e^{-u}\). \(\square\)

The count uses the finest grid in the block of the edge \(e_i\), not the finest grid of the whole construction. Lemma 2.1 of Dyadic grids and windows on a tree bounds its resolution in terms of the window length \(r\) alone; this is what lets the exponential gain \(e^{-p(M-1)r/2}\) beat the count \(D_{M,q}(r)\), which grows like \(rq^{-2r}\).

2. The choice of the parameters

Fix \(p\in(0,1)\). Choose, in this order:

  1. an integer \(M\ge2\) with \[ \begin{aligned}\alpha&:=\frac{p(M-1)}2\\&>2\log\frac1q ;\end{aligned}\tag{2.1} \]
  2. an integer \(d\ge1\) with \((1-2^{1-M})^d<p\);
  3. with \(K\) the number of edges of the ordered \(M\)-ary tree of height \(d\), an integer \(g\ge1\) with \(4Kc\,q^{g+1}<p\);
  4. an integer \(r_*\ge1\) such that for every integer \(r\ge r_*\) \[ D_{M,q}(r)\,e^{-\alpha r}<p .\tag{2.2} \]

The last choice is possible because \(D_{M,q}(r)e^{-\alpha r}\) is at most a constant times \(r\,q^{-2r}e^{-\alpha r}=r\,e^{-(\alpha-2\log(1/q))r}\), which tends to \(0\) by (2.1). Build the windows of Dyadic grids and windows on a tree with these \(M\), \(d\), \(g\) and \(r_0=r_*\); then every \(r_h\ge r_*\). Build the random set \(B\) of Random routing and independent tests with these windows and the given \(p\).

3. Misses at stable centers

Lemma 3.1 (stable centers; [OpenAI-ES, Section 5]). For every stable center \(x\), \[ \begin{aligned}&\mathbb P\Bigl(\exists t\in[1,2]\ \forall n\in\mathcal N:\\&\qquad x+tq^n\notin B\Bigr)\le2p .\end{aligned}\tag{3.1} \]

Proof. Split \(\Omega\) according to the atoms of the exposure of \(x\). The atoms on which the route makes no default choice have total probability \((1-2^{1-M})^d<p\), by Lemma 2.1 of Random routing and independent tests and the choice of \(d\). On an atom with a default choice, Lemma 1.1, \(r_h\ge r_*\) and (2.2) bound the conditional probability of the event by \(p\). Summing, the probability is less than \(p+p\). \(\square\)

The event in (3.1) quantifies over all \(t\in[1,2]\), but \(\Omega\) is finite, so it is an event like any other subset of \(\Omega\).

4. Every center: the proof of Proposition 3.1

Proof of Proposition 3.1 of Measure-universal sets and periodic hitting sets ([OpenAI-ES, Section 5]).

Open enlargement. For each outcome, \(B\) is \(1\)-periodic and in each period a union of finitely many half-open cells \([u,v)\); replacing each by \((u-\varepsilon,v)\), with \(\varepsilon\) times the number of cells at most \(p\), gives an open \(1\)-periodic set \(B^+\supseteq B\) with \[ \rho(B^+)\le\rho(B)+p . \]

The missed centers. For each outcome put \[ \begin{aligned}R=\bigl\{&x\in\mathbb R:\ \exists t\in[1,2]\\&\forall n\in\mathcal N,\\&x+tq^n\notin B^+\bigr\}.\end{aligned}\tag{4.1} \] \(R\) is \(1\)-periodic because \(B^+\) is. It is closed: let \(x_k\in R\) converge to \(x\), with \(t_k\in[1,2]\) witnessing \(x_k\in R\). By the Bolzano–Weierstrass theorem a subsequence of \((t_k)\) converges to some \(t\in[1,2]\). For each \(n\in\mathcal N\), the points \(x_k+t_kq^n\) along this subsequence converge to \(x+tq^n\) and lie in the closed set \(\mathbb R\setminus B^+\), so \(x+tq^n\notin B^+\). Hence \(x\in R\).

Expected sizes. If \(x\) is stable and \(x\in R\), then the event of (3.1) occurs, since \(B\subseteq B^+\). With Lemma 3.2 of Dyadic grids and windows on a tree and the choice of \(g\), \(\rho(\mathbb R\setminus\mathcal S)\le4Kcq^{g+1}<p\). Since \(\Omega\) is finite, \[ \begin{aligned}\mathbb E\rho(R)&=\int_0^1\mathbb P(x\in R)\,dx\\&\le2p\,\rho(\mathcal S)+\rho(\mathbb R\setminus\mathcal S)<3p,\end{aligned} \] and \(\mathbb E\rho(B^+)\le\mathbb E\rho(B)+p=2p\) by Lemma 1.1 of Random routing and independent tests. So some outcome satisfies \[ \rho(B^+)+\rho(R)<5p . \] Fix such an outcome.

Covering the missed centers. For \(k\ge1\) let \(V_k=\{x:\operatorname{dist}(x,R)<1/k\}\), an open \(1\)-periodic set; if \(R\) is empty, put \(V_k=\varnothing\). The sets \(V_k\cap[0,1)\) decrease, and their intersection is \(R\cap[0,1)\) because \(R\) is closed. By continuity from above, \(\rho(V_k)\to\rho(R)\); choose \(k\) with \(\rho(V_k)\le\rho(R)+p\) and put \(V=V_k\) and \[ H=B^+\cup V . \] Then \(H\) is open and \(1\)-periodic, and \(\rho(H)\le\rho(B^+)+\rho(R)+p<6p\).

Every center is hit. Let \(x\in\mathbb R\) and \(t\in[1,2]\). If \(x\notin R\), there is \(n\in\mathcal N\) with \(x+tq^n\in B^+\subseteq H\). If \(x\in R\), then \(x\in V\), which is open; since \(x+tq^n\to x\) as \(n\to\infty\), we get \(x+tq^n\in V\subseteq H\) for all large \(n\). These indices need not lie in \(\mathcal N\), and (3.1) of Measure-universal sets and periodic hitting sets allows every \(n\ge1\). \(\square\)

With Section 3 of Measure-universal sets and periodic hitting sets, Theorem 2.2 there is proved: for every \(q\in(0,1)\) and \(\eta\in(0,1)\), a compact subset of \([0,1]\) of measure greater than \(1-\eta\) contains no affine copy of \(\{q^n:n\ge1\}\). Everything used is proved in this course, apart from basic facts of Lebesgue measure and real analysis from the core courses.

The construction is explicit but very large: the branching number \(M\) is of the order of \(\log(1/q)/p\), and the height \(d\) of the order of \(2^{M}\log(1/p)\) (Exercise 5.3).

5. Exercises

5.1. Show that for \(\alpha=2\log(1/q)\) the quantity \(D_{M,q}(r)e^{-\alpha r}\) tends to infinity as \(r\to\infty\). Which inequality in the proof needs the strict inequality (2.1)?

5.2. Show that \(R\) in (4.1) is closed by a second argument: write \(R\) as the projection to the first coordinate of a closed subset of \(\mathbb R\times[1,2]\), and show that such projections are closed.

5.3. Let \(q=\frac12\) and \(p=\frac1{10}\). Find the least \(M\) allowed by (2.1) and the least \(d\) allowed by the second choice in Section 2.

5.4. In the proof of Proposition 3.1, why does one not take \(H=B^+\)? Describe where the indices outside \(\mathcal N\) are used.

5.5. Show that \(H\) in the proof satisfies the stronger property: for every \(x\in\mathbb R\) and \(t\in[1,2]\), either some \(n\in\mathcal N\) has \(x+tq^n\in B^+\), or all sufficiently large \(n\) have \(x+tq^n\in H\).

6. Solutions

5.1. For \(\alpha=2\log(1/q)\), \(e^{-\alpha r}=q^{2r}\), so \(D_{M,q}(r)e^{-\alpha r}\ge4c(M-1)r\to\infty\). The strict inequality in (2.1) is needed for the existence of \(r_*\) in (2.2).

5.2. \(R\) is the projection to the first coordinate of the set \(Z\) of pairs \((x,t)\in\mathbb R\times[1,2]\) with \(x+tq^n\notin B^+\) for all \(n\in\mathcal N\), which is closed as an intersection of preimages of the closed set \(\mathbb R\setminus B^+\) under continuous maps. If \(x_k\to x\) with \((x_k,t_k)\in Z\), a subsequence of \(t_k\) converges to some \(t\in[1,2]\) by the Bolzano–Weierstrass theorem, and \((x,t)\in Z\) because \(Z\) is closed; so the projection is closed. The compactness of \([1,2]\) is essential: the projection of the closed set \(\{(x,t):xt=1\}\subseteq\mathbb R\times\mathbb R\) is \(\mathbb R\setminus\{0\}\), not closed.

5.3. (2.1) asks \((M-1)/20>2\log2\approx1.386\), so \(M-1\ge28\) and \(M=29\). Then \(d\) must satisfy \((1-2^{-28})^d<\frac1{10}\), that is \(d>\log10/(-\log(1-2^{-28}))\), approximately \(6.18\times10^8\); the least such \(d\) is \(618{,}095{,}479\).

5.4. The set \(R\) of missed centers can be nonempty: the random set gives every center only a probability of being hit, and unstable centers are not controlled at all. The open set \(V\supseteq R\) of small density repairs them, and the points \(x+tq^n\) that it contains have large indices \(n\), which need not belong to any window.

5.5. This is the case distinction of the last paragraph of the proof: if \(x\notin R\), some \(n\in\mathcal N\) works with \(B^+\); if \(x\in R\subseteq V\), all large \(n\) work with \(V\subseteq H\).

References