Abstract

In this paper, we study the Erdős distinct distances problem for Cartesian product sets in the setting of arbitrary finite fields. More precisely, let Fq be an arbitrary finite field and A be a set in Fq. Suppose |A∩(aG)|≤|G|1/2 for any subfield G and a∈F∗q, then ∣∣ΔFq(A2)∣∣=∣∣(A−A)2+(A−A)2∣∣≫∣∣A∣∣1+121. Using the same method, we also obtain some results on sum–product type problems.