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Statement

Notation as on Theorem 1.1: $x_n(\lambda)=\lfloor\lambda2^n\rfloor$ for real $\lambda>0$.

Theorem 1.2 (p. 3). There is an infinite sequence $S=\{n_k:k\ge1\}$ with $n_1=2$ and

$$2^{\frac1{14}(n_{k-1}+2k-7)}\le n_k\le2^{27(n_{k-1}+2k+6)}, \qquad(1.4)$$

such that the set $\Sigma(S)$ of all real $\lambda>0$ for which every integer $x_n(\lambda)$ with $n\in S$ has a ternary expansion omitting the digit $2$ is uncountable.

The growth condition (1.4) is printed without a range for $k$; since it involves $n_{k-1}$, it is read for $k\ge2$. Every $\lambda\in\Sigma(S)$ has infinitely many $n$ with $(\lfloor\lambda2^n\rfloor)_3$ omitting the digit $2$, so lies in the truncated real exceptional set of Theorem 1.3. The paper adds without proof (p. 3) that (1.4) gives $\#\{n_k:1\le n_k\le X\}\ge\log_*(X)-4$ for $X\ge2$, where $\log_*(X)$ is the number of iterations of the logarithm starting at $X$ needed to get a value smaller than $1$, and hence $N_\lambda(X)\ge\log_*(X)-4$ for every $\lambda\in\Sigma(S)$, displays (1.5) and (1.6).

Source. Theorem 1.2, 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 remarks after it were read clause by clause on the page image. The proof (pp. 13--16) was read for its structure only. Nothing here is independently reviewed.

Proof pointer

Pp. 13--16. The proof builds exponents $m_k=l_0+l_1+\cdots+l_k$, with $l_k$ chosen so that the fractional part of $l_k\log_32$ is positive and very small, which makes $2^{l_k}$ a ternary $1$ followed by a long run of zeros. It then builds a Cantor-type set $\tilde\Sigma\subset[1,2]$ of reals $\sum_k d_k2^{-m_k}$, branching at least twice at every level, for which each $\lfloor\lambda2^{m_k}\rfloor$ omits the digit $1$. An integer omitting the digit $1$ is even and is twice an integer omitting the digit $2$, so $S=\{m_k-1\}$ works. The bounds (1.4) come from the continued fraction of $\log_32$ via Lemma 2.2 (p. 10).

Dependencies

Lemma 2.2 (p. 10) of the same paper, the Diophantine bound for $\log_32$.

Bears on

• Problem 406: the theorem concerns perturbed starting values $\lambda$, not the powers of $2$ themselves, and says nothing about $\lambda=1$. It shows that the analogue of the problem's finiteness fails for uncountably many $\lambda$ in the truncated real system.