More on the $h$-critical numbers of finite abelian groups

zbMATH Open1399.11065arXiv1611.07269MaRDI QIDQ3133268

Publication date: 13 February 2018

Abstract: For a finite abelian group

G

, a nonempty subset

A

of

G

, and a positive integer

h

, we let

h A

denote the

h

-fold sumset of

A

; that is,

h A

is the collection of sums of

h

not-necessarily-distinct elements of

A

. Furthermore, for a positive integer

s

, we set

[0, s] A = c u p_{h = 0}^{s} h A

. We say that

A

is a generating set of

G

if there is a positive integer

s

for which

[0, s] A = G

. The

h

-critical number

c h i (G, h)

of

G

is defined as the smallest positive integer

m

for which

h A = G

holds for every

m

-subset

A

of

G

; similarly,

c h i (G, [0, s])

is the smallest positive integer

m

for which

[0, s] A = G

holds for every

m

-subset

A

of

G

. We define

w i d e h a t c h i (G, h)

as the smallest positive integer

m

for which

h A = G

holds for every generating

m

-subset

A

of

G

;

w i d e h a t c h i (G, [0, s])

is defined similarly. The value of

c h i (G, h)

has been determined by this author for all

G

and

h

, and

w i d e h a t c h i (G, [0, s])

was introduced and resolved for some special cases by Klopsch and Lev. Here we determine the remaining two quantities in all cases.

Full work available at URL: https://arxiv.org/abs/1611.07269

zbMATH Keywords

finite abelian groups $h$-critical numbers

Mathematics Subject Classification ID

Other combinatorial number theory (11B75) Finite abelian groups (20K01) Arithmetic progressions (11B25) Extremal combinatorics (05D99) Inverse problems of additive number theory, including sumsets (11P70)