On a theorem of Hildebrand

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Carsten Dietzel, On a theorem of Hildebrand, Moscow Journal of Combinatorics and Number Theory 8 (2019), 189–191, DOI 10.2140/moscow.2019.8.189. The edition read is arXiv:1810.01518v1, 2 October 2018, three pages; the journal edition has not been compared.

Theorem 4 (p. 2) concerns any multiplicative subgroup $A\le\mathbb Q_{>0}$ of finite index. It supplies positive integers $b_1,b_2,\ldots$ such that, for every nonempty finite set $F$ of indices,

$$s=\sum_{i\in F}b_i\quad\Longrightarrow\quad s,s+1\in A.$$

Thus the integer pair starts contain an IP-set. The proof uses Hindman's finite-unions theorem and a sequence with divisibility between ordered block sums. Dividing by the first selected block makes that block equal to $1$. Finite sums use distinct blocks; this construction does not supply a second copy of the normalized first block to obtain $s+2$.

Read status: claims checked. The full TeX was read, and Theorem 4, its proof and Comment 1 were compared with the page images on pp. 2–3. The proof was read for structure; the external Hindman theorem was not checked. Definition 1 prints a decreasing sequence, whereas Lemma 1 constructs an increasing one. The source's printed wording is retained.

Bears on. The pair theorem strengthens the known pair case of Problem 436. Comment 1 explicitly leaves the odd-index triple question open beyond index $3$.

The arXiv record, read on 2026-10-17, gives the non-exclusive distribution license. No file of this source is held because no license on record permits redistribution.