We prove a Chernoff-like large deviation bound on the sum of non-independent random variables that have the following dependence structure. The variables $Y_1,\ldots,Y_r$ are arbitrary Boolean functions of independent random variables $X_1,\ldots,X_m$, modulo a restriction that every $X_i$ influences at most $k$ of the variables $Y_1,\ldots,Y_r$.

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A $(k,\epsilon)$-biased sample space is a distribution over $\{0,1\}^n$ that $\epsilon$-fools every nonempty linear test of size at most $k$. Since they were introduced by Naor and Naor [SIAM J. Computing, 1993], these sample spaces have become a central notion in theoretical computer science with a variety of applications.

When ... more >>>

We show that sparse affine-invariant linear properties over arbitrary finite fields are locally testable with a constant number of queries. Given a finite field ${\mathbb{F}}_q$ and an extension field ${\mathbb{F}}_{q^n}$, a property is a set of functions mapping ${\mathbb{F}}_{q^n}$ to ${\mathbb{F}}_q$. The property is said to be affine-invariant if it ... more >>>