It is shown that if (X, ||.||_X) is a Banach space with Rademacher cotype q
then for every integer n there exists an even integer m< n^{1+1/q}$ such that
for every f:Z_m^n --> X we have \sum_{j=1}^n \Avg_x [ ||f(x+ (m/2) e_j)-f(x)
||_X^q ] < C m^q \Avg_{\e,x} [ ||f(x+\e)-f(x) ||_X^q ], where the expectations
are with respect to uniformly chosen x\in Z_m^n and \e\in \{-1,0,1\}^n, and all
the implied constants may depend only on q and the Rademacher cotype q constant
of X. This improves the bound of m< n^{2+\frac{1}{q}} from [Mendel, Naor 2008].
The proof of the above inequality is based on a "smoothing and approximation"
procedure which simplifies the proof of the metric characterization of
Rademacher cotype of [Mendel, Naor 2008]. We also show that any such "smoothing
and approximation" approach to metric cotype inequalities must require m>
n^{(1/2)+(1/q)}.