$C-(k, \ell)$-Sum-Free Sets

zbMATH Open1487.11026arXiv2001.00327MaRDI QIDQ5864869

Publication date: 10 June 2022

Abstract: The Minkowski sum of two subsets

A

and

B

of a finite abelian group

G

is defined as all pairwise sums of elements of

A

and

B

:

A + B = a + b : a i n A, b i n B

. The largest size of a

(k, e l l)

-sum-free set in

G

has been of interest for many years and in the case

G = m a t h b b Z / n m a t h b b Z

has recently been computed by Bajnok and Matzke. Motivated by sum-free sets of the torus, Kravitz introduces the noisy Minkowski sum of two sets, which can be thought of as discrete evaluations of these continuous sumsets. That is, given a noise set

C

, the noisy Minkowski sum is defined as

A +_{C} B = A + B + C

. We give bounds on the maximum size of a

(k, e l l)

-sum-free subset of

m a t h b b Z / n m a t h b b Z

under this new sum, for

C

equal to an arithmetic progression with common difference relatively prime to

n

and for any two element set

C

.

Full work available at URL: https://arxiv.org/abs/2001.00327

zbMATH Keywords

Minkowski sum

Mathematics Subject Classification ID

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