Abstract
For a nonempty subset
B
of a group
G
, let
μ
G
,
B
(
n
,
c
)
$\mu _{G,B}(n,c)$
mu Subscript upper G comma upper B Baseline left parenthesis n comma c right parenthesis
be the least possible value of
max
∅
≠
S
⊆
B
,
|
S
|
≤
c
|
A
S
|
$\max _{\varnothing \ne S\subseteq B,\ |S|\le c}|AS|$
max Underscript empty set not equals upper S subset of or equal to upper B comma StartAbsoluteValue upper S EndAbsoluteValue less than or equals c Endscripts StartAbsoluteValue upper A upper S EndAbsoluteValue
over all
n
-element subsets
A
⊆
G
$A\subseteq G$
upper A subset of or equal to upper G
. Given
b
0
∈
B
$b_0\in B$
b 0 element of upper B
, set
D
=
b
0
−
1
B
$D=b_0^{-1}B$
upper D equals b 0 Superscript negative 1 Baseline upper B
and
H
=
⟨
D
⟩
$H=\langle D\rangle $
upper H equals left angle bracket upper D right angle bracket
. We prove that
μ
G
,
B
(
n
,
c
)
$\mu _{G,B}(n,c)$
mu Subscript upper G comma upper B Baseline left parenthesis n comma c right parenthesis
is determined exactly by the based affine-core
(
H
,
D
)
$(H,D)$
left parenthesis upper H comma upper D right parenthesis
. The proof uses a simultaneous block compression that controls all admissible multiplier sets at once. As a consequence of the exact determination,
μ
Z
,
λ
B
(
n
,
c
)
=
μ
Z
,
B
(
n
,
c
)
$\mu _{\mathbb Z,\lambda B}(n,c)=\mu _{\mathbb Z,B}(n,c)$
mu Subscript double struck upper Z comma lamda upper B Baseline left parenthesis n comma c right parenthesis equals mu Subscript double struck upper Z comma upper B Baseline left parenthesis n comma c right parenthesis
for every nonempty
B
⊆
Z
$B\subseteq \mathbb Z$
upper B subset of or equal to double struck upper Z
and every nonzero integer
λ
$\lambda $
lamda
. This proves, in the stronger form of an exact identity, the natural nonzero-dilation version of a conjecture of Bollobás
et al.
[‘Large sumsets from small subsets’,
Israel J. Math.
268
(1) (2025), 253–314].