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Small sample spaces cannot fool low degree polynomials Ido Ben-Eliezer
 

Summary: Small sample spaces cannot fool low degree polynomials
Noga Alon
Ido Ben-Eliezer
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

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics