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Statement

Setting (p. 141). For a finite nondecreasing sequence of positive integers $a_1,\ldots,a_k$ and an integer $s\ge0$, $\Sigma(\mathrm{Pow}(\{a_1,\ldots,a_k\}),s)$ is the set of all sums of terms $a_i^r$ over distinct pairs $(i,r)$ with $r\ge s$ and $1\le i\le k$, the empty sum included, and $P_{\{a_1,\ldots,a_k\}}(x)$ is the number of its elements $n$ (for $s=0$) with $n\le x$. For $\{3,4\}$ and $s=0$ the set consists of the numbers $a+b$ with $a$ a sum of distinct powers $3^i$ ($i\ge0$) and $b$ a sum of distinct powers $4^j$ ($j\ge0$).

Theorem 4 (p. 146, quoted). "Let $P_{\{3,4\}}(x)$ be the counting function of $\Sigma(\mathrm{Pow}(\{3,4\}),0)$. We have $P_{\{3,4\}}(x)\gg x^{0.97777}$."

That is, there is a constant $C>0$ with $P_{\{3,4\}}(x)\ge Cx^{0.97777}$ for all sufficiently large $x$. The proof gives the exponent as $\gamma=1-(\tau/2)(1/\log3-1/\log4)$, which the paper states exceeds $0.97777$, where

$$\tau=\frac{1}{\log(4/3)}\int_0^{\log(4/3)}\bigl(-\log k(e^u)\bigr)\,du\simeq0.2353664$$

with $k$ the function of the paper's Definition 1 (see Lemma 3); the value of $\tau$ is computed numerically, and the paper points to its code at http://github.com/m-f-h/SumPow34 (p. 147). The paper states that the theorem improves Melfi's bound $P_{\{3,4\}}(x)\gg x^{0.965}$ (G. Melfi, An additive problem about powers of fixed integers, Rend. Circ. Mat. Palermo (2) 50 (2001), 239--246), and its closing remarks (p. 148) say that an iteration over three or more cycles appears out of reach of present computation, so it appears very difficult to improve the estimate with these techniques.

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 setting on p. 141, the cycles $B_n$ on p. 146, Theorem 4 on p. 146 and its proof on pp. 146--147. The edition read is identified on the source card.

Read depth. Claims checked: the setting and the statement were read clause by clause on the printed pages. The proof was read but not checked step by step, and the numerical value of $\tau$ was not recomputed. Nothing here is independently reviewed.

Proof pointer

Pp. 146--147. The increasing sequence of powers of $3$ and $4$ is cut, at each pair of consecutive powers of $3$, into cycles $B_n$ of $7$ or $9$ terms, starting at $3^{r_n}$ with $3^{r_n}<4^{\ell_n}<3^{r_n+1}$, and $c_n=4^{\ell_n}/3^{r_n}\in(1,4/3)$. If $P_{\{3,4\}}(x)\ge ax$ for all $x\le d_n$, where $d_n$ is the largest element of the set below $3^{r_n}$, then the elements up to $d_{n+2}$ lie in a union of $2^{18}$ (or $2^{16}$) translated copies of $[0,d_n]$ indexed by the sums of the powers in $B_n\cup B_{n+1}$, and this gives $P_{\{3,4\}}(x)\ge a\,k(c_n)(1-\varepsilon)x$ for all $x\le d_{n+2}$ and large $n$. Iterating over pairs of cycles multiplies these factors. Since $\log4/\log3$ is irrational, $\log c_n$ is uniformly distributed in $[0,\log(4/3)]$, so the average of $\log k(c_n)$ tends to $-\tau$; the index $n$ of the cycle reached at $x$ is $(1/\log3-1/\log4)\log x+\kappa$ with $\lvert\kappa\rvert<5$, and the $n/2$ factors give the exponent $\gamma$.

Dependencies

The function $k$ and its continuity off $3^9/4^7$ (Definition 1 and Lemma 2, p. 142); the minimum value of $k$ is Lemma 3, which the proof does not use directly.

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 $\Sigma(\mathrm{Pow}(\{3,4\}),0)$, so Theorem 4 is a lower bound $\lvert(A+B)\cap[0,x]\rvert\gg x^{0.97777}$ for its counting function. A bound of order $x^{0.97777}$ does not decide whether the lower density is positive, and the paper does not settle it.