Hasler 2024 sums distinct powers 3 4

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lemma_3: Hasler and Melfi's computation that the auxiliary function k of their Definition 1, a minimal average density of a union of intervals built from powers of 3 and 4, has minimum value 1015/1458 on [1,4/3], attained at 1.

proposition_5: Hasler and Melfi's upper bound 1015/1458, about 0.69616, for the lower asymptotic density of the set of sums of distinct powers of 3 and distinct powers of 4.

theorem_4: Hasler and Melfi's lower bound for the counting function of the set of sums of distinct powers of 3 and distinct powers of 4: it is at least a positive constant times x to the power 0.97777, improving Melfi's exponent 0.965.

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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. Received 21 January 2024, revised 31 May 2024. MSC2020 primary 11A67, secondary 11B37.

The copy read for this card is the publisher's (Mathematical Sciences Publishers) PDF, eleven PDF pages: the journal's article cover page (PDF p. 1), the eight printed pages 141--148 (PDF pp. 2--9; PDF p. $n$ is printed p. $n+139$), and the journal's masthead and issue table of contents (PDF pp. 10--11), with a text layer. Provenance: downloaded in the survey of September 2026; the survey record identifies the source by the DOI 10.2140/cnt.2024.13.141 (https://doi.org/10.2140/cnt.2024.13.141), which the first page also prints, and the download URL itself was not recorded; 689,544 bytes. The copy prints "© 2024 MSP (Mathematical Sciences Publishers)." on printed p. 141 and "© 2024 Mathematical Sciences Publishers" on the journal's masthead (PDF p. 10), every other right reserved.

Read status: claims checked for Lemma 3, Theorem 4 and Proposition 5, whose statements were read clause by clause on the printed pages; their proofs were read but not verified; the problem page does not yet consume any statement from this source. Result pages: Lemma 3, Theorem 4, Proposition 5.

Contents

• Setting (p. 141): $\Sigma(\mathrm{Pow}(\{a_1,\dots,a_k\}),s)$ is the set of sums of terms $a_i^r$ over distinct pairs $(i,r)$ with $r\ge s$, including the empty sum, and $P_{\{a_1,\dots,a_k\}}(x)$ counts the elements $\le x$ of the $s=0$ set. For $\{3,4\}$ and $s=0$ this set is exactly the sumset $A+B$ of the problem, and by Melfi 2001, Proposition 1 (the problem's [Me01]), $P_{{3,4},1}(x)\le P_{{3,4}}(x)\le 4P_{{3,4},1}(x)$, so the $s=0$ and $s=1$ versions have positive lower density together. The paper attributes the positive-density question to Burr, Erdős, Graham and Li 1996 ([BEGL96]) and the conjecture of a positive answer to Erdős 1997 ([Er97]), whom its abstract credits with conjecturing in 1996 that the $s=1$ set has positive asymptotic density; the previous best lower bound was $P_{\{3,4\}}(x)\gg x^{0.965}$ (Melfi 2001).
• Section 2 (pp. 142--145): the function $k(c)$ on $[1,4/3]$ (Definition 1, p. 142), the least value of $\frac1x\int_0^x\mathbb 1_{A_c}$ over $x\le\max A_c$ for a set $A_c$ defined from unions of intervals $[\alpha+\beta c,\alpha+\beta c+1/2+c/3]$ over $\alpha\in\Sigma(1,3,\dots,3^9)$, $\beta\in\Sigma(1,4,\dots,4^7)$ (or $3^8$, $4^6$ for $c\ge3^9/4^7$); Lemma 2 (continuity off $3^9/4^7$ and piecewise form $A+Bc$ or $p+q/c$); Lemma 3 (p. 145): $\min k=k(1)=1015/1458\approx0.69616$, computed from the fact that the only positive integers $\le243$ outside $\Sigma(\mathrm{Pow}(\{3,4\}),0)$ are $62,63,143,144$ and the $36$ integers from $207$ to $242$.
• Theorem 4 (p. 146; proof pp. 146--147): $P_{\{3,4\}}(x)\gg x^{0.97777}$. Statement read clause by clause. The proof iterates over two consecutive "cycles" of the merged sequence of powers of $3$ and $4$, uses $k(c_n)$ with $c_n=4^{\ell_n}/3^{r_n}$, and the uniform distribution of $\log c_n$ in $[0,\log(4/3)]$; the exponent is $1-(\tau/2)(1/\log3-1/\log4)>0.97777$ with $\tau=\frac{1}{\log(4/3)}\int_0^{\log(4/3)}(-\log k(e^u))\,du\approx0.2353664$ computed numerically (code at http://github.com/m-f-h/SumPow34).
• Proposition 5 (p. 147; proof pp. 147--148): $\liminf_{x\to\infty}P_{\{3,4\}}(x)/x\le k(1)=1015/1458\approx0.69616$. Statement read clause by clause.
• Section 4 (p. 148): the two-cycle iteration is at the limit of present computation, so the method is unlikely to improve the exponent; Open Question 6 asks whether $\Sigma(\mathrm{Pow}(\{a_1,\dots,a_k\}),1)$ has positive asymptotic density whenever $\sum1/\log a_i>1/\log2$ and the $a_i$ are pairwise coprime, generalizing the Erdős conjecture. The paper notes that, by Melfi 2001, for $k>1$ the condition $\sum1/\log a_i>1/\log2$ is necessary for positive upper asymptotic density of the $s=1$ set, and that it is not sufficient: Melfi 2001's example $\{3,9,81\}$ satisfies it and has zero asymptotic density.

Compiled scope

The whole article (printed pp. 141--148) was read on the printed pages. The statements of Theorem 4, Proposition 5 and Lemma 3, with Definition 1 and Lemma 2 on which they rest, were checked clause by clause; the proofs, which rest on computations the paper reports (Table 1, the numerical value of $\tau$), were read but not verified and the computations were not repeated. Nothing here is independently reviewed.

Bears on. #125: the problem's $A+B$ is $\Sigma(\mathrm{Pow}(\{3,4\}),0)$. Theorem 4 is the lower bound $P_{\{3,4\}}(x)\gg x^{0.97777}$ for its counting function, which the paper states improves Melfi's $x^{0.965}$ of 2001, and Proposition 5, with the value of Lemma 3, is the upper bound $1015/1458$ for its lower density. Neither bound decides whether the lower density is positive, and the paper does not settle it.

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