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Statement

The $3$-adic exceptional set (1.10) (p. 4) is

$$\mathcal E(\mathbb Z_3)=\{\lambda\in\mathbb Z_3:\text{infinitely many 3-adic expansions }\lambda2^n\text{ omit the digit }2\},$$

and for $k\ge1$ the approximating sets (1.11) (p. 4) are

$$\mathcal E^{(k)}(\mathbb Z_3)=\{\lambda\in\mathbb Z_3:\text{at least }k\text{ values of }\lambda2^n\text{ omit the digit }2\}.$$

They are nested, $\mathcal E^{(1)}(\mathbb Z_3)\supset\mathcal E^{(2)}(\mathbb Z_3)\supset\cdots$, and each contains $\mathcal E(\mathbb Z_3)$. Hausdorff dimension is taken for the $3$-adic metric; put $\alpha_0=\log_32\approx0.63092$.

Theorem 1.5 (pp. 4--5).

1. $\dim_H(\mathcal E^{(1)}(\mathbb Z_3))=\alpha_0\approx0.63092$, display (1.12).
2. $\frac12\log_3(2)\le\dim_H(\mathcal E^{(2)}(\mathbb Z_3))\le\frac12$, display (1.13).
3. $\mathcal E^{(3)}(\mathbb Z_3)$ has positive Hausdorff dimension, with $\frac16\log_32\le\dim_H(\mathcal E^{(3)}(\mathbb Z_3))\le\dim_H(\mathcal E^{(2)}(\mathbb Z_3))$, display (1.14).

The paper states (p. 5) that it is not clear whether $\dim_H(\mathcal E^{(k)}(\mathbb Z_3))>0$ for all $k\ge1$, and remarks after the proof (p. 22) that its method gives no positive lower bound for any $k\ge4$, since it uses the known solutions $2^0,2^2,2^8$ of Erdős's problem. Its Conjecture B (p. 5) asserts that $\mathcal E(\mathbb Z_3)$ has Hausdorff dimension zero, and it states (p. 5) that Erdős's conjecture is equivalent to $1\notin\mathcal E(\mathbb Z_3)$.

Source. Theorem 1.5, pp. 4--5, 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 (1.10) and (1.11) were read clause by clause on the page images, and the proof (pp. 20--21) was followed except for the dimension of the $3$-adic Cantor set, which it takes from a comparison with the real Cantor set (p. 20). Nothing here is independently reviewed.

Proof pointer

Pp. 20--21. $\mathcal E^{(k)}(\mathbb Z_3)$ is the countable union of the sets $\mathcal C(2^{m_1},\ldots,2^{m_k})$ of $\lambda$ with every $\lambda2^{m_j}$ omitting the digit $2$, so its dimension is the supremum of theirs. Part (1): each $\mathcal C(2^m)$ is a scaled copy of the $3$-adic Cantor set, of dimension $\log_32$. Part (2): $\mathcal C(2^{m_1},2^{m_2})$ is a scaled copy of $\mathcal C(1,2^{m_2-m_1})$, whose dimension is at most $\frac12$ by Theorem 1.6; for the lower bound, $4=(11)_3$ and the digit-pair set with blocks $00$, $01$ lies in $\mathcal C(1,4)$. Part (3): $4=(11)_3$ and $256=(100111)_3$, and the set with six-digit blocks $000000$, $000001$ lies in $\mathcal C(1,4,256)$. The proof's display for part (3) prints $\mathcal E^{(2)}(\mathbb Z_3)$ where the union is over triples, a misprint for $\mathcal E^{(3)}(\mathbb Z_3)$ (p. 21).

Dependencies

Theorem 1.6 for the upper bound in part (2).

Bears on

• Problem 406: the paper states that the problem's assertion is equivalent to $1\notin\mathcal E(\mathbb Z_3)$. Since $2^0$, $2^2$ and $2^8$ omit the digit $2$, the point $1$ lies in $\mathcal E^{(3)}(\mathbb Z_3)$ (an observation of this page); the theorem bounds the size of these approximating sets and does not decide whether $1$ lies in $\mathcal E(\mathbb Z_3)$. The upper bound $\frac12$ in part (2) is also an upper bound for $\dim_H(\mathcal E(\mathbb Z_3))$. Later lower bounds on $\mathcal E^{(2)}(\mathbb Z_3)$ and $\mathcal E^{(3)}(\mathbb Z_3)$ are on Abram and Lagarias, Theorem 5.2.