Let
M
M
be a lineally convex hypersurface of
C
n
\mathbb C^n
of finite type,
0
∈
M
0\in M
. Then there exist non-trivial smooth CR functions on
M
M
that are flat at
0
0
, i.e. whose Taylor expansion about
0
0
vanishes identically. Our aim is to characterize the rate at which flat CR functions can decrease without vanishing identically. As it turns out, non-trivial CR functions cannot decay arbitrarily fast, and a possible way of expressing the critical rate is by comparison with a suitable exponential of the modulus of a local peak function.