On the largest product-free subsets of the alternating groups
Name
222_2024_Article_1273.pdf
Size
1.36 MB
Format
Adobe PDF
Checksum (MD5)
9e832efe5344db391852c128c785478d
Author(s)
Keevash, Peter
Lifshitz, Noam
Minzer, Dor
Date Issued
May 29, 2024
Journal
Inventiones mathematicae
Publisher
Springer Science and Business Media LLC
Citation
Keevash, P., Lifshitz, N. & Minzer, D. On the largest product-free subsets of the alternating groups. Invent. math. (2024).
Version
Final published version
Abstract
Abstract
A subset
A
$A$
of a group
G
$G$
is called product-free if there is no solution to
a
=
b
c
$a=bc$
with
a
,
b
,
c
$a,b,c$
all in
A
$A$
. It is easy to see that the largest product-free subset of the symmetric group
S
n
$S_{n}$
is obtained by taking the set of all odd permutations, i.e.
S
n
∖
A
n
$S_{n} \backslash A_{n}$
, where
A
n
$A_{n}$
is the alternating group. In 1985 Babai and Sós (Eur. J. Comb. 6(2):101–114, 1985) conjectured that the group
A
n
$A_{n}$
also contains a product-free set of constant density. This conjecture was refuted by Gowers (whose result was subsequently improved by Eberhard), still leaving the long-standing problem of determining the largest product-free subset of
A
n
$A_{n}$
wide open. We solve this problem for large
n
$n$
, showing that the maximum size is achieved by the previously conjectured extremal examples, namely families of the form
{
π
:
π
(
x
)
∈
I
,
π
(
I
)
∩
I
=
∅
}
$\left \{ \pi :\,\pi (x)\in I, \pi (I)\cap I=\varnothing \right \} $
and their inverses. Moreover, we show that the maximum size is only achieved by these extremal examples, and we have stability: any product-free subset of
A
n
$A_{n}$
of nearly maximum size is structurally close to an extremal example. Our proof uses a combination of tools from Combinatorics and Non-abelian Fourier Analysis, including a crucial new ingredient exploiting some recent theory developed by Filmus, Kindler, Lifshitz and Minzer for global hypercontractivity on the symmetric group.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00222-024-01273-1
