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Statement

The truncated real exceptional set is defined in (1.7) (p. 3) as

$$\mathcal E_T(\mathbb R_+)=\{\lambda>0:\text{infinitely many ternary expansions }(\lfloor\lambda2^n\rfloor)_3\text{ omit the digit }2\}.$$

Theorem 1.3 (p. 3). The set $\mathcal E_T(\mathbb R_+)$ has Hausdorff dimension

$$\dim_H(\mathcal E_T(\mathbb R_+))=\log_3(2)=\frac{\log2}{\log3}\approx0.63092,$$

and it has nonzero $\log_3(2)$-dimensional Hausdorff measure.

The paper distinguishes this set from the untruncated real exceptional set $\mathcal E(\mathbb R_+)$ of (1.8) (p. 3), defined with the full ternary expansions $(\lambda2^n)_3$ of the real numbers $\lambda2^n$, which it states may even be empty and for which its Conjecture A (p. 4) asserts Hausdorff dimension zero. The paper states (p. 3) that Erdős's conjecture is equivalent to $1\notin\mathcal E(\mathbb R_+)$.

Source. Theorem 1.3, p. 3, 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 definition (1.7) were read clause by clause on the page image. The proof (pp. 16--19) was read for its structure only. Nothing here is independently reviewed.

Proof pointer

Pp. 16--19. Upper bound: for $\lambda\in[1/M,M]$ and each $j$, the integers $\lfloor\lambda2^j\rfloor$ omitting the digit $2$ number at most $4M2^{j\alpha_0}$, each fixing $\lambda$ to an interval of length $2^{-j}$; summing over $j\ge n$ covers $\mathcal E_T(\mathbb R_+)\cap[1/M,M]$ with total $(\alpha_0+\epsilon)$-mass tending to $0$. Lower bound: the set $\tilde\Sigma$ built in the proof of Theorem 1.2 lies in $\mathcal E_T(\mathbb R_+)$, and a Cantor-set mass argument adapted from Falconer shows its $\alpha_0$-dimensional Hausdorff measure exceeds $1/16$, display (2.27) (p. 17).

Dependencies

The construction in the proof of Theorem 1.2.

Bears on

• Problem 406: with $\lambda=1$ the integers $\lfloor\lambda2^n\rfloor$ are the powers $2^n$, so the problem asks whether $1$ lies outside $\mathcal E_T(\mathbb R_+)$. The theorem measures the size of this set and does not decide whether $1$ belongs to it; the paper offers it (p. 3) as an indication of why deciding membership for a particular $\lambda$ may be hard.