Covers of abelian groups by cosets

../

corollary_1_1: For an m-cover of an arbitrary group by k left cosets, a point a covered exactly m times and any abelian subgroup K, the number of cosets missing a whose subgroup does not contain K is at most k − m and at least f([K : K ∩ H_a]), where H_a is the intersection of the subgroups whose cosets contain a.

theorem_1_3: For an m-cover of an abelian group by k left cosets and a point a covered exactly m times, the index N_a of the intersection of the subgroups whose cosets contain a satisfies N_a ≤ 2^(k−m) and k ≥ m + f(N_a), with f the Mycielski function; an irredundant coset a_tG_t gives the same bounds for [G:G_t], and these are best possible.

theorem_2_1: For an m-cover of the integers by k residue classes and an integer a covered exactly m times, k ≥ m + f(N_a) with N_a the least common multiple of the moduli of the classes containing a, and for each prime p a weighted count of the classes I(p) is at least ord_p(N_a)(p − 1).

---

Günter Lettl and Zhi-Wei Sun, On covers of abelian groups by cosets, Acta Arithmetica 131 (2008), no. 4, 341–350, doi:10.4064/aa131-4-3.

The copy read for this card is the publisher facsimile, the official IMPAN download (10 physical pages, printed journal pages 341–350). The original arXiv:math/0411144v2 PDF is a distinct 10-page version; its first-page header prints the same journal citation while its own folios are 1–10. The statement digest gives paired publisher printed-page and arXiv internal-folio locators. The official Acta Arithmetica record identifies the publisher version, and the arXiv record identifies the preprint. The statement digest was read against both versions. The publisher facsimile prints "© Instytut Matematyczny PAN, 2008" on its first page (printed p. 341); the publisher's issue listing labels the article "Free download under CC-BY license", a Creative Commons Attribution license with no version named (https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/131/4, read 2026-10-02), and that named license on the publisher's page decides over the printed line; the site footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the site, not the article. For the arXiv v2 PDF, the arXiv record carries no license field, so arXiv's assumed license applies (arXiv:math/0411144), every other right reserved.

Located results

Here $f$ is the Mycielski function $f(n)=\sum_{p\mid n}\operatorname{ord}_p(n)(p-1)$ (Definition 1.1, p. 341), and, for a subnormal subgroup $H$ of finite index in $G$, $d(G,H)$ is the sum of $[H_i:H_{i-1}]-1$ along any composition series from $H$ to $G$ (Definition 1.2, p. 342).

Theorem 1.2 (p. 342; arXiv folio 3), credited to Korec and Sun. If left cosets $a_1G_1,\ldots,a_kG_k$ of subnormal subgroups of a group $G$ cover every element of $G$ exactly $m$ times, then $[G:\bigcap_sG_s]<\infty$ and

$$k\ge m+d\Bigl(G,\bigcap_sG_s\Bigr)\ge m+f\Bigl(\Bigl[G:\bigcap_sG_s\Bigr]\Bigr),$$

the bound $m+d(G,\bigcap_sG_s)$ being best possible. The paper adds (p. 342) that under the same hypotheses Sun [S04] showed that the indices $[G:G_s]$ are not pairwise distinct when $k>1$.

Theorem 1.3 (p. 343; arXiv folio 4), the main result. Let $\{a_sG_s\}_{s=1}^k$ be an $m$-cover of an abelian group $G$ by left cosets, so that every element lies in at least $m$ of them. For any $a\in G$ lying in exactly $m$ of them, the index $N_a$ of the intersection of the $G_s$ with $a\in a_sG_s$ satisfies

$$N_a\le2^{k-m}\qquad\text{and}\qquad k\ge m+f(N_a).$$

In particular, if $\{a_sG_s\}_{s\ne t}$ is not an $m$-cover, then $[G:G_t]\le2^{k-m}$ and $k\ge m+f([G:G_t])$, and these bounds are best possible. The case $m=1$ gives the Gao–Geroldinger conjecture for every finite abelian group (p. 343). Its page is Theorem 1.3.

Corollary 1.1 (p. 344; arXiv folios 4–5). For an $m$-cover of an arbitrary group, a point $a$ covered exactly $m$ times and any abelian subgroup $K$, the cosets missing $a$ whose subgroups do not contain $K$ number at most $k-m$ and at least $f([K:K\cap\bigcap_{a\in a_sG_s}G_s])$; its page is Corollary 1.1.

Theorem 2.1 (p. 346; arXiv folio 7). For an $m$-cover of $\mathbb Z$ by $k$ residue classes and an integer $a$ covered exactly $m$ times, $k\ge m+f(N_a)$ with $N_a$ the least common multiple of the moduli of the classes containing $a$, together with a prime-by-prime refinement (2.4); its page is Theorem 2.1.

Bears on

• Problem 274: the paper reports on p. 342 the result of Sun [S04] that an exact cover by $k>1$ cosets of subnormal subgroups never has pairwise distinct indices, so the problem's answer is no for such covers; the paper's own results concern $m$-covers and do not address the problem.
• Problem 1189: the case $G=\mathbb Z$, $m=1$ of Theorem 1.3, or Theorem 2.1, gives $n_t\le2^{k-1}$ for every modulus of an irreducible covering set of size $k$, a deduction recorded on the result pages and not a statement of the paper; Remark 1.2 (p. 343) credits that case of Theorem 1.3 to Znám (1975).

The source also belongs to the covering-system context of [[covering_systems/sun_2004_herzog_schonheim_conjecture_uniform_covers/_index|Sun's uniform-cover work]].

Only the edition under an open license is held; the source's other editions are not, since no license on record permits their redistribution, and the card cites the edition it names above.