Let Fq be a finite field of order q. We prove that if d ≥ 2 is even and E ⊂ F d q with |E| ≥ 9q d 2 then 2000 Mathematics Subject Classification. 11T24, 52C17.

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Let f (t 1 , . . . , tn) be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function D f (X), the number of integers of absolute value up to X represented by f . When f is isotropic or n is at least 3, we show that there is a δ(f ) ∈ Q ∩ (0, 1) such that D f (X) ∼ δ(f )X and call δ(f ) the density of f . We consider the inverse problem of which densities arise. Our main technical tool is a Near Hasse Principle: a quadratic form may fail to represent infinitely many integers that it locally represents, but this set of exceptions has density 0 within the set of locally represented integers.

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