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Statement

For a real $\lambda>0$ put $x_n(\lambda)=\lfloor\lambda2^n\rfloor$ for $n\ge0$, the truncated real dynamical system of (1.1) (p. 2), and write $(k)_3$ for the ternary expansion of an integer $k$.

Theorem 1.1 (p. 2). For each $\lambda>0$, the count

$$N_\lambda(X)=\#\{n:1\le n\le X\text{ and }(\lfloor\lambda2^n\rfloor)_3\text{ omits the digit }2\}$$

satisfies $N_\lambda(X)\le25X^{0.9725}$ for all sufficiently large $X\ge n_0(\lambda)$.

The threshold $n_0(\lambda)$ depends on $\lambda$ and is not made explicit in the statement; the constants $25$ and $0.9725$ do not depend on $\lambda$. For $\lambda=1$ the integers are the powers $2^n$ themselves, so the theorem bounds the count of Problem 406's exponents $1\le n\le X$. For that value Narkiewicz's earlier bound $N_1(X)\le1.62X^{\alpha_0}$, with $\alpha_0=\log_32\approx0.63092$, which the paper records (p. 1), is stronger.

Source. Theorem 1.1, p. 2, 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 was read clause by clause on the page image. The proof (pp. 11--13) was read for its structure only. Nothing here is independently reviewed.

Proof pointer

Pp. 11--13. Writing $w_n=\log_3(\lambda2^n)\pmod 1$, the leading $k$ ternary digits of $x_n(\lambda)$ are fixed by which of $2\cdot3^{k-1}$ intervals $J(b)$ contains $w_n$, and $w_n$ is an orbit of rotation by $\log_32$. The three-distance theorem (Lemma 2.1, p. 9) puts at most six points of each block of $q_{l-1}-1$ consecutive $w_n$ in any $J(b)$, where $q_{l-1}<X\le q_l$ are continued-fraction denominators of $\log_32$; only $2^{k-1}$ leading blocks omit the digit $2$. The growth bound $q_l\le1200\,q_{l-1}^{13.3}$ (Lemma 2.2, p. 10), derived from a linear-forms-in-logarithms estimate of Simons and de Weger, converts the count into a power of $X$.

Dependencies

Lemma 2.1 (p. 9), the three-distance theorem for an irrational rotation, and Lemma 2.2 (p. 10), the Diophantine bound for $\log_32$, both of the same paper; Lemma 2.2 rests on results cited from Simons and de Weger and from Rhin.

Bears on

• Problem 406: with $\lambda=1$ the theorem says that at most $25X^{0.9725}$ exponents $1\le n\le X$ give a power $2^n$ with only the digits $0$ and $1$ in base $3$, for all large $X$. This is a density bound, weaker than Narkiewicz's bound for that case, and does not decide whether there are finitely many such powers.