Let G be a group acting on a set
Ω
\Omega
and k a non-negative integer. A subset (finite or infinite)
A
⊆
Ω
A \subseteq \Omega
is called k-quasi-invariant if
|
A
g
∖
A
|
≤
k
|{A^g}\backslash A| \leq k
for every
g
∈
G
g \in G
. It is shown that if A is k-quasi-invariant for
k
≥
1
k \geq 1
, then there exists an invariant subset
Γ
⊆
Ω
\Gamma \subseteq \Omega
such that
|
A
△
Γ
|
>
2
e
k
⌈
(
ln
2
k
)
⌉
|A\vartriangle \Gamma | > 2ek\left \lceil {(\ln 2k)} \right \rceil
. Information about G-orbit intersections with A is obtained. In particular, the number m of G-orbits which have non-empty intersection with A, but are not contained in A, is at most
2
k
−
1
2k - 1
. Certain other bounds on
|
A
△
Γ
|
|A\vartriangle \Gamma |
, in terms of both m and k, are also obtained.