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Statement

Setting (p. 141). $P_{\{3,4\}}(x)$ counts the elements $n\le x$ of $\Sigma(\mathrm{Pow}(\{3,4\}),0)$, the set of sums of distinct powers $3^i$ and distinct powers $4^j$ with $i,j\ge0$, the empty sum included (see Theorem 4 for the general notation). $k$ is the function on $D=[1,4/3]$ of the paper's Definition 1 (p. 142; see Lemma 3).

Proposition 5 (p. 147, quoted). "Let $P_{\{3,4\}}(x)$ be the counting function of $\Sigma(\mathrm{Pow}(\{3,4\}),0)$. We have"

$$\liminf_{x\to\infty}\frac{P_{\{3,4\}}(x)}{x}\le k(1)=\frac{1015}{1458}\simeq0.69616.\qquad(7)$$

The value $k(1)=1015/1458$ is Lemma 3. The proposition bounds the lower density from above; it gives no positive lower bound for it.

Source. M. F. Hasler and G. Melfi, On sums of distinct powers of 3 and 4, Combinatorics and Number Theory 13 (2024), no. 2, 141--148, doi:10.2140/cnt.2024.13.141: the bound announced on p. 142, Proposition 5 on p. 147 and its proof on pp. 147--148. The edition read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the printed page. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pp. 147--148. For each $\varepsilon>0$ it suffices to find arbitrarily large $x$ with $P_{\{3,4\}}(x)/x<k(1)+\varepsilon$. Because $\log4/\log3$ is irrational, for every $\delta>0$ there are infinitely many $m,l$ with $3^m<4^l<(1+\delta)3^m$. Along such $m$ the paper takes $x=3^{m+5}-1$ and, from the structure developed for Theorem 4, bounds the limit of $P_{\{3,4\}}(x)/x$ by $k(1+\delta)$. Continuity of $k$ near $1$ (Lemma 2) makes $k(1+\delta)<k(1)+\varepsilon$ for small $\delta$, and Lemma 3 supplies the value of $k(1)$.

Dependencies

Lemma 2 (p. 142, continuity of $k$ on $D\setminus\{3^9/4^7\}$) and Lemma 3 of the same paper.

Bears on

• Problem 125: the problem asks whether $A+B$ has positive lower density, where $A$ and $B$ are the integers with only digits $0,1$ in base $3$ and in base $4$; that sumset is $\Sigma(\mathrm{Pow}(\{3,4\}),0)$. Proposition 5 shows its lower density is at most $1015/1458$. An upper bound below $1$ does not decide whether the lower density is positive, and the paper does not settle it.