Integer distances in vector spaces over finite fields

<p>The Erdős-Anning Theorem states that an integer distance set in the Euclidean plane must have all of its points on a single line or is finite. However, this is not true if we consider area sets. That is, if <span class="math inline">\((x_1,y_1)\)</span> and <span class="math inline">\((x_2,y_2)\)</span> are any two vectors contained in the integer lattice, then the area of the parallelogram determined by the two vectors is an integer, showing that the points do not have to lie on a line. We prove a finite field version of these results for <span class="math inline">\(d=2\)</span> and <span class="math inline">\(d=3\)</span>, showing that if <span class="math inline">\(E \subset \Bbb{F}_q^d, q=p^2\)</span>, where <span class="math inline">\(p\)</span> is an odd prime and the distance set of <span class="math inline">\(E\)</span> is <span class="math inline">\(\Bbb{F}_p\)</span>, then the size of <span class="math inline">\(E\)</span> is at most <span class="math inline">\(p^d\)</span>. Furthermore, we prove that if the area set of <span class="math inline">\(E\)</span> is a subset of <span class="math inline">\(\Bbb{F}_p\)</span>, then the size of <span class="math inline">\(E\)</span> is at most <span class="math inline">\(p^2\)</span> in two dimensions.</p>