A combinatorial problem for vector spaces over finite fields

Erdos Distance Problem in Vector Spaces over Finite Fields

The classical Erdos distance conjecture says that the number of distance determined by N points in Rd, d ≥ 2, is at least CN 2 d − . We shall discuss this problem in vector spaces over finite fields where some interesting number theory comes into play. A connection with the continuous analog of the Erdos distance conjecture, the Falconer distance conjecture will be also be established. Universi...

Extension and averaging operators in vector spaces over finite fields

We study the mapping properties of both extension operators and averaging operators associated with the algebraic variety S = {x ∈ Fq : Q(x) = 0} where Q(x) is a nondegenerate quadratic form over the finite field Fq. Carbery, Stones, and Wright [3] established a method to deduce L − L estimates for averaging operators over a general variety. As a core result in this paper, we give the sharp est...

Canonical Quantization of Symplectic Vector Spaces over Finite Fields

In this paper an affirmati5e answer is given to a question of D. Kazhdan on the existence of a canonical Hilbert spaces attached to symplectic vector spaces over finite fields. This result suggest a solution to a discrete analogue of a well known problem in geometric quantization where a naive canonical Hilbert space does not exist. As a result, a canonical model for the Weil representation of ...

Harmonic Analysis on Vector Spaces over Finite Fields

• (i) G = ZN = Z/NZ = {0, 1, 2, ....., N − 1} with addition modulo N . For 0 ≤ n ≤ N − 1 let γn : G → S, γn(m) = exp(2πimn/N). Then {γ0, ....., γN−1} is a complete list of the characters so that ZN is isomorphic to ZN . An example of a primitive N ’th root of unity is ω := exp 2πi/N . • (ii) G = T = R/Z; for n ∈ Z let γn : G→ S, γn(x) = exp(2πinx). Then G∗ = {γn : n ∈ Z} so that G∗ is isomorphi...

Extension Theorems in Vector Spaces over Finite Fields