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Statement

Notation (p. 2). $\mathbb N$ is the set of positive integers; for sequences, $x_n=O(y_n)$ means $|x_n|\le C|y_n|$ for every $n\in\mathbb N$ with some constant $C\in(0,\infty)$, and a vector is $O(y_n)$ when each coordinate is. Vectors of $\mathbb R^3$ are written as columns, and $\mathbb e_1,\mathbb e_2,\mathbb e_3$ is the standard basis (p. 8).

Lemma 2 (p. 8). There exist a matrix $M\in\mathrm{GL}(3,\mathbb R)$, mutually disjoint finite sets $S_1,S_2,S_3,T_1,T_2,T_3\subset\mathbb N$, and constants $c_1,c_2,c_3\in(0,\infty)$ such that, for $1\le j\le3$,

$$\Bigl(\sum_{a\in S_j}-\sum_{a\in T_j}\Bigr)M \begin{pmatrix}1/(an)\\1/(an+1)\\1/(an+2)\end{pmatrix} =\frac{c_j}{n^j}\,\mathbb e_j+O\Bigl(\frac1{n^4}\Bigr). \qquad(3.1)$$

Explicit data (proof, p. 9). The proof takes

$$M=\begin{pmatrix}1&0&0\\3&-4&1\\1&-2&1\end{pmatrix},\qquad\det M=-2, \qquad(3.2)$$

$S_1=\{45,72,144,160,432,480\}$, $T_1=\{48,60,120,720,1440,4320\}$, $S_2=11\cdot\{16,20,240\}$, $T_2=11\cdot\{15,24,120\}$, $S_3=7\cdot\{10,30,60\}$, $T_3=7\cdot\{12,15\}$, and obtains $c_1=1/180$, $c_2=1/348480$, $c_3=1/1029000$. The factors $7$ and $11$ only make the six sets mutually disjoint. The paper remarks (p. 9) that its use of the lemma in the proof of Theorem 1 does not need the sets to be finite, only to have finitely many prime factors in all.

Check of the data (an observation of this page, not of the paper). With exact rational arithmetic, $\sum_{S_j}a^{-i}-\sum_{T_j}a^{-i}$ for $i=1,2,3$ vanishes for $i\ne j$ and equals $1/180$, $1/696960$, $1/2058000$ for $i=j=1,2,3$; with the rows of $M$, which send $(1/(an),1/(an+1),1/(an+2))$ to $(1/(an),\,2/(an)^2,\,2/(an)^3)+O(1/n^4)$, these give the printed $c_1$, $c_2=2/696960$ and $c_3=2/2058000$. The 23 listed elements are distinct.

Source. V. Kovač, On the set of points represented by harmonic subseries, arXiv:2405.07681v3 (12 September 2024); Amer. Math. Monthly 132 (2025), 895--911: Lemma 2 in Section 3 on p. 8, its proof on p. 9 (arXiv v3 pagination). The edition read is identified on the source card.

Read depth. Claims checked: the statement and the explicit data were read clause by clause on the printed pages, and the power-sum identities behind $c_1,c_2,c_3$ were recomputed as stated above. Nothing here is independently reviewed.

Proof pointer

Page 9. Expanding $1/(an+k-1)$ in powers of $1/(an)$ up to an $O(1/n^4)$ error, the matrix $M$ makes the second coordinate start at $2/(an)^2$ and the third at $2/(an)^3$. The lemma then reduces to finding disjoint sets whose reciprocal power sums agree in the two powers other than $j$ and differ in power $j$; the paper supplies six elementary identities among unit fractions and their squares and cubes, verified by computer.

Dependencies

None beyond elementary expansions and the stated identities.

Bears on

• Problem 268: only as the arithmetic step in the proof of Theorem 1, whose page states the relation; the lemma alone says nothing about the set in the problem.