A two-parameter finite field Erdős-Falconer distance problem

2017/02/07 by Birklbauer, Philipp, Iosevich, Alex
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1702.02126

Abstract

We study the following two-parameter variant of the Erd\H os-Falconer distance problem. Given E,F ⊂ \Bbb F_q^k+l, l ≥ k ≥ 2, the k+l-dimensional vector space over the finite field with q elements, let B_k,l(E,F) be given by \(\Vert x'-y'\Vert, \Vert x"-y" \Vert): x=(x',x") ∈ E, y=(y',y") ∈ F; x',y' ∈ \Bbb F_q^k, x",y" ∈ \Bbb F_q^l \. We prove that if |E||F| ≥ C q^k+2l+1, then B_k,l(E,F)=\Bbb F_q × \Bbb F_q. Furthermore this result is sharp if k is odd. For the case of l=k=2 and q a prime with q ≡ 3 \mod 4 we get that for every positive C there is c such that if |E||F|gt;C q^6+(2)/(3), then |B_2,2(E,F)|gt; c q².

Related

• On the two-parameter Erdős-Falconer distance problem over finite fields
• On the Erdős-Falconer distance problem for two sets of different size in vector spaces over finite fields
• The Erdős-Falconer distance problem between arbitrary sets and k-coordinatable sets in finite fields
• Erdos distance problem in vector spaces over finite fields
• The Erdös-Falconer distance problem on the unit sphere in vector spaces over finite fields