3 M ar 2 01 4 Results on the Erdős-Falconer distance problem in Z dq for odd q

The Erdős-Falconer distance problem in Zq asks one to show that if E ⊂ Zq is of sufficiently large cardinality, then ∆(E) := {(x1 − y1) + · · · + (xd − yd) : x, y ∈ E} satisfies ∆(E) = Zq. Here, Zq is the set of integers modulo q, and Z d q = Zq × · · · × Zq is the free module of rank d over Zq. We extend known results in two directions. Previous results were known only in the setting q = p, where p is an odd prime, and as such only showed that all units were obtained in the distance set. We remove the constriction that q is a power of a prime, and despite this, shows that the distance set of E contains all of Zq whenever E is of sufficiently large cardinality. 1 Background and Results A large portion of geometric combinatorics asks one to show that if a set is sufficiently large, then it exhibits some specific type of geometric structure. One of the best known such results is the Erdős-distance problem. Let f(n) be the minimum number of Euclidean distances determined by any set of n points in R. The classical Erdősdistance problem asks one to show that there exists a constant c such that f(n) ≥ { cn d = 2 cn d ≥ 3 As usual, we write X = o(Y ) if X/Y tends to zero as some common parameter of X and Y tends to infinity. The case d = 2 was recently resolved by Guth-Katz ([9]) with an ingenious application of the polynomial method. The conjecture remains open for d ≥ 3. See [6, 8, 13, 16, 17] and the references therein for more background and a thorough treatise of the problem. A continuous analog of the Erdős-distance problem is due to Falconer ([15]). For a compact set E ⊂ [0, 1], let ∆(E) = {|x−y| : x, y ∈ E} be the set of pairwise Euclidean distances in E. Falconer showed ([7]) that if dimH(E) > d+1 2 , then L1(∆(E)) > 0, where dimH(·) denotes Hausdorff dimension and L1(·) denotes 1-dimensional Lebesgue measure. Further, he constructed a compact set E ⊂ [0, 1] with dimH(E) = d2 such 1 that ∆(E) did not have positive Lebesgue measure. This led him to conjecture that if E ⊂ [0, 1] is compact and dimH(E) > d2 , then L1(∆(E)) > 0. The best know results in this direction are due to Wolff (in the plane) and Erdoğan (for d ≥ 3) who showed that L1(∆(E)) > 0 whenever E ⊂ [0, 1] is compact with dimH(E) > d2 + 1 3 ([5, 18]). A finite field analogue of the distance problem was first considered by BourgainKatz-Tao ([2]). Let Fq denote the d-dimensional vector space over the finite field with q elements. For E ⊂ Fq , define ∆(E) = {‖x− y‖ : x, y ∈ E} ⊂ Fq where ‖x− y‖ = (x− y)(x − y) = (x1 − y1) + · · ·+ (xd − yd). Clearly, ‖·‖ is not a norm, though this notion of distance is preserved under orthogonal transformations. More precisely, let Od(Fq) denote the set of d×d orthogonal matrices with entries in Fq. One can readily check that for O ∈ Od(Fq), we have ‖x‖ = ‖Ox‖. Once a suitable notion of distance has been established in Fq , the problem proceeds just as before. Problem 1.1 (Erdős-Falconer Distance Problem). Find the minimal exponent α such that if q is odd then there exists a constant C (independent of q) so that for any set E ⊂ Fq of cardinality |E| ≥ Cq, we have ∆(E) = Fq. Note that when q = p, then Fp2 contains a subfield isomorphic to Fp, and hence, there exists a set E ⊂ Fq such that |E| = q and |∆(E)| = √ q. This shows that the exponent α from Problem 1.1 cannot be less than d2 in general. Remark 1.2. The methods used to attack the Erdős-Falconer distance problem are of a much different flavor when q is even. For example, in F2, we can construct a large set that contains only 0 in its distance set as follows. Let E ⊂ F2 be the set of vectors which have an even number of nonzero components. Then, ‖x− y‖ = 0 in F2 since ‖x− y‖ = (x1 − y1) + . (xd − yd) = x1 + y1 + · · ·+ xd + yd. Note also that E has cardinality