 
Summary: Small sample spaces cannot fool low degree polynomials
Noga Alon
Ido BenEliezer
Michael Krivelevich
Abstract
A distribution D on a set S ZN
p fools polynomials of degree at most d in N variables
over Zp if for any such polynomial P, the distribution of P(x) when x is chosen according
to D differs from the distribution when x is chosen uniformly by at most in the 1 norm.
Distributions of this type generalize the notion of biased spaces and have been studied in
several recent papers. We establish tight bounds on the minimum possible size of the support
S of such a distribution, showing that any such S satisfies
S c1 ·
( N
2d )d
· log p
2 log (1
)
+ p .
This is nearly optimal as there is such an S of size at most
