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Statement

Let $\Sigma_{3,\bar2}$ be the set of $\lambda\in\mathbb Z_3$ whose $3$-adic expansion omits the digit $2$, display (1.15) (p. 5). For integers $1\le M_1<\cdots<M_k$ the multiplicative intersection set $\mathcal C(M_1,\ldots,M_k)$ is the set of $\lambda\in\mathbb Z_3$ with $(M_j\lambda)_3$ omitting the digit $2$ for $1\le j\le k$, display (1.16) (p. 5), that is, the intersection of the sets $\frac1{M_j}\Sigma_{3,\bar2}$. The second line of (1.16) prints this as a union, a misprint: the set-builder line and Theorem 1.6 itself use the intersection.

Theorem 1.6 (p. 6). Let $M$ be a positive integer which is not a power of $3$. Then

$$\dim_H(\mathcal C(1,M))=\dim_H\Bigl(\Sigma_{3,\bar2}\cap\tfrac1M\Sigma_{3,\bar2}\Bigr)\le\frac12. \qquad(1.18)$$

The paper does not know whether the bound is sharp, and states without proof (p. 6) that $\dim_H(\mathcal C(1,7))=\log_3\frac{1+\sqrt5}2\approx0.438$. For $M$ a power of $3$, $\mathcal C(1,M)=\Sigma_{3,\bar2}$ has dimension $\log_32$ (p. 23).

Source. Theorem 1.6, p. 6, of Jeffrey C. Lagarias, Ternary expansions of powers of 2, J. Lond. Math. Soc. (2) 79 (2009), no. 3, 562--588; labels and pages are those of the arXiv:math/0512006v4 edition (11 July 2008) identified on the source card.

Read depth. Claims checked: the statement and the definitions (1.15) and (1.16) were read clause by clause on the page images. The proof (pp. 23--24) was read for its structure only. Nothing here is independently reviewed.

Proof pointer

Pp. 23--24, with Lemma 4.1 and the reduction before it (p. 22). Since $\mathcal C(1,3^jM)=\mathcal C(1,M)$, one may take $M$ prime to $3$, and Lemma 4.1 (p. 22) gives $\mathcal C(1,M)=\{0\}$ when $M\equiv2\pmod 3$. For $M\equiv1\pmod3$ with lowest nonzero digit above the units digit in position $m$, digits of $\lambda$ are paired $m$ apart; in each pair at most three of the four digit choices keep the corresponding digit of $M\lambda$ away from $2$ (Claim 1, p. 23). Hence at most about $3^{r/2}$ classes modulo $3^r$ meet $\mathcal C(1,M)$ (Claim 2, p. 24), which gives the bound $\frac12$.

Dependencies

Lemma 4.1 (p. 22) of the same paper.

Bears on

• Problem 406: through Theorem 1.5, applied with $M=2^{m_2-m_1}$, it gives the upper bound $\frac12$ for the dimension of $\mathcal E^{(2)}(\mathbb Z_3)$ and hence of the $3$-adic exceptional set $\mathcal E(\mathbb Z_3)$, whose non-membership of $1$ the paper states is equivalent to the problem's assertion. It does not decide the problem.