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Source. Lemma 1, p. 3, 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.

Read depth. Claims checked: the statement and its setting were read clause by clause on the print. Nothing here is independently reviewed.

Statement

Setting (pp. 2--3). Inside the lower-bound part of the proof of the Theorem, $p>1$ is an odd integer, $A_p$ is the increasing sequence of integers $2^xp^y$ with $x,y\ge0$, and $\delta>0$, $\epsilon>0$ are fixed small. For each integer $j\ge0$ put $x_j=(1+\epsilon)^j$ and let $X_j$ be the number of elements of $A_p$ in the interval $[x_j,(p-\delta)x_j)$.

Lemma 1 (p. 3). There is a constant $c_p$ such that

$$X_j<\frac{\log x_j\,\log(p-\delta)}{\log2\,\log p}+c_p \qquad\text{for all }j\ge0.$$

Proof pointer

The paper gives no proof; it takes the lemma from the discussion in Lecture 5 of G. H. Hardy, Ramanujan: Twelve lectures on subjects suggested by his life and work (Chelsea, 1940), its reference [6]. In the proof of the Theorem (p. 3) the number of short sums whose smallest term lies in $[x_j,x_{j+1})$ is at most $2^{X_j}$, and the lemma gives $2^{X_j}<2^{c_p}x_j^{\log(p-\delta)/\log p}$.

Dependencies

Hardy's Lecture 5, as cited above.

Bears on

• Problem 845: through part 3 of the Theorem only; at $p=3$ the lemma is the counting input showing that for $1<C<3$ almost all integers are not sums of distinct 3-smooth numbers with largest term below $C$ times the smallest.