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Statement

Setting (p. 142). $D=[1,4/3]$, and $\Sigma(x,y,z,\ldots)$ denotes $\{0,x\}+\{0,y\}+\{0,z\}+\cdots$, the set of subset sums of the listed numbers. For $c\in D$, $A_c$ is the union of the closed intervals

$$\Bigl[\alpha+\beta c,\ \alpha+\beta c+\tfrac12+\tfrac c3\Bigr]$$

over $\alpha\in\Sigma(1,3,9,\ldots,3^9)$ and $\beta\in\Sigma(1,4,16,\ldots,4^7)$ when $1\le c<3^9/4^7$ (the paper's (1)), and over $\alpha\in\Sigma(1,3,9,\ldots,3^8)$ and $\beta\in\Sigma(1,4,16,\ldots,4^6)$ when $3^9/4^7\le c\le4/3$ (the paper's (2)).

Definition 1 (p. 142). $k:D\to\mathbb R$ is

$$k(c)=\min\Bigl\{\frac1x\int_0^x\mathbb 1_{A_c}(t)\,dt:\ x\in[0,\max A_c]\Bigr\}, \qquad(3)$$

the smallest proportion of $[0,x]$ covered by $A_c$, over $x$ up to the right end of $A_c$. (The print writes the range as $[0,\max A_c]$, where the quotient is undefined at $x=0$; Figure 1's legend, p. 143, writes $0<x<\max A_c$.)

Lemma 2 (p. 142) states that $k$ is continuous on $D\setminus\{3^9/4^7\}$ and piecewise of the form $A+Bc$ or $p+q/c$ with $A,B,p,q\in\mathbb Q$; the paper computes that $k$ is discontinuous at $3^9/4^7$ (p. 143) and lists its pieces in Table 1 (p. 145).

Lemma 3 (p. 145, quoted). "Let $k:D\to\mathbb R$ be defined as above. We have"

$$\min\{k(c):c\in D\}=k(1)=\frac1{243}\int_0^{243}\mathbb 1_{A_1}(t)\,dt=\frac{1015}{1458}\simeq0.69616.$$

The proof also states that $k(c)>k(1)$ for every $c\in(1,4/3]$, so the minimum is attained only at $c=1$; that part rests on the complete computation of $k$ (Lemma 2, Table 1 and Figure 1).

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, Definition 1 and Lemma 2 on p. 142, the proof of Lemma 2 on pp. 142--144, Table 1 and Lemma 3 with its proof on p. 145. The edition read is identified on the source card.

Read depth. Claims checked: the setting, Definition 1 and the statements of Lemmas 2 and 3 were read clause by clause on the printed pages. The proofs were read but not checked step by step, and the computation of $k$ over $D$ was not repeated. Nothing here is independently reviewed.

Proof pointer

P. 145. At $c=1$ the intervals have length $5/6$ and left ends at the integers $\alpha+\beta$; the paper states that the only positive integers up to $243$ outside $\Sigma(\mathrm{Pow}(\{3,4\}),0)$ are $62,63,143,144$ and the $36$ integers from $207$ to $242$, and the minimizing $x$ is $243$, so $k(1)=\frac56\cdot\frac{243-2-2-36}{243}=\frac{1015}{1458}$. Near $c=1$ the function is affine, $k(c)-k(1)=(c-1)\cdot12440/729$ for $1\le c\le513/512$, and the computation of $k$ on the rest of $D$ shows $k(c)>k(1)$ there.

Dependencies

Lemma 2 and Table 1 of the same paper. The lemma is used in the proof of Proposition 5.

Bears on

• Problem 125: through Proposition 5, the value $k(1)=1015/1458$ is the paper's upper bound for the lower density of $A+B=\Sigma(\mathrm{Pow}(\{3,4\}),0)$. The lemma by itself does not decide whether that lower density is positive.