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Source. The unnumbered Theorem on p. 2 of Wouter van Doorn and Anneroos R. F. Everts, Smooth sums with small spacings, arXiv:2511.04585v1 (6 November 2025), the edition identified on the source card; its proof runs from p. 2 to p. 7.

Read depth. Claims checked: the statement and the definitions it uses were read clause by clause on the print, and the proof was read for its structure. Nothing here is independently reviewed.

Statement

Setting (p. 2). Let $p>1$ be an odd integer and let $A_p=(a_1,a_2,\ldots)$ be the increasing sequence of all integers $2^{x}p^{y}$ with $x,y\ge0$ integers. With $\log_2$ the logarithm to base 2, put $f_0(x)=x$, $f_k(x)=\max\bigl(1,\lfloor\log_2f_{k-1}(x)\rfloor\bigr)$ for $k\ge1$, and $F(x)=\prod_{k\ge0}f_k(x)$.

Theorem (p. 2).

1. For every odd integer $p>1$ there is a constant $C_p$ such that every positive integer $n$ can be written as $n=b_1+b_2+\cdots+b_r$ with every $b_i\in A_p$ and $b_1<b_2<\cdots<b_r<C_pb_1$.
2. In general one may take $C_p=\tfrac12F(4p)$. If $p-1$ is a power of two one may take $C_p=2p$, and if $p+1$ is a power of two one may take $C_p=2(p+1)$.
3. No constant smaller than $p$ can replace $C_p$.

The case $p=3$ (abstract, p. 1, and p. 2). Here $p-1=2$, so $C_3=6$: every positive integer $n$ is a sum $n=b_1+\cdots+b_r$ of distinct 3-smooth integers with $1\le b_1<b_2<\cdots<b_r<6b_1$. Part 3 says no constant below $3$ works for the 3-smooth integers.

What the proof of part 3 establishes (pp. 2--3) is a density statement: call a sum of distinct elements of $A_p$ with $b_1<\cdots<b_r<Cb_1$ short; then for every constant $C$ with $1<C<p$, almost all positive integers are not short sums.

Proof pointer

Lower bound (pp. 2--3). Fix $1<C<p$ and small $\delta,\epsilon>0$ with $C(1+\epsilon)<p-\delta$. Split by the interval $[(1+\epsilon)^j,(1+\epsilon)^{j+1})$ that holds $b_1$; every summand of a short sum then lies in a window of ratio $p-\delta$, and Lemma 1 (p. 3) bounds the number of elements of $A_p$ in such a window. Summing the resulting bounds over $j$ shows that the number of short sums with all summands at most $N$ is at most $2^{c_p}(L+1)N^{\log(p-\delta)/\log p}$ with $L=\lfloor\log N/\log(1+\epsilon)\rfloor$, which is small compared with $N$.

Existence (pp. 3--7). Lemma 2 (p. 4) supplies a set $S\subset A_p\setminus\{1\}$ whose subset sums cover $\lvert S\rvert+1$ consecutive integers starting at some $M_0\le p$. Starting from a representation of $n$ with coefficients $M_0$ or $M_0+1$ on the smallest elements of $A_p$ and a binary expansion of the remainder, a variant of the "midgame" procedure of Blecksmith, McCallum and Selfridge (the paper's reference [5]) lowers each coefficient in turn to $0$ or $1$ by raising coefficients of larger elements of $A_p$; Lemma 3 (p. 6) bounds the coefficients interval by interval, so all surviving summands lie in $[a_m,C_pa_m)$ for $C_p$ a product of factors $u_k$ fixed by $S$ and $M_0$. The special values $2p$ and $2(p+1)$ come from the choices $S=\{2,p-1,p\}$ (a multiset when $p=3$) and $S=\{2,p,p+1\}$ (p. 7); the general bound $\tfrac12F(4p)$ is the computation at the end of p. 7.

Section 3 (p. 8) shows by example that multisets can lower $C_p$ further for some $p$, and leaves open whether a constant $c$ with $C_p<cp$ for all odd $p>1$ exists.

Dependencies

Lemma 1, which the paper draws from Lecture 5 of Hardy's Ramanujan (its reference [6]); Lemmas 2 and 3 of the paper; the procedure of Blecksmith, McCallum and Selfridge, 3-smooth representations of integers, Amer. Math. Monthly 105 (1998).

Bears on

• Problem 845: the problem asks, for a constant $C$, whether the integers $b_1+\cdots+b_t$ with distinct $b_i=2^{k_i}3^{l_i}$ and $b_t\le Cb_1$ have density $0$. The case $p=3$ gives every positive integer such a sum with $b_t<6b_1$, so for every $C\ge6$ the set is all positive integers. Part 3, at $p=3$, shows that for every $1<C<3$ almost all integers are not such sums with $b_t<Cb_1$. The paper decides nothing for $3\le C<6$.