The analog of the Erdös distance problem in finite fields

In this paper, we give a proof of the result of Iosevich and Rudnev [Erdös distance problem in vector spaces over finite fields, Trans. Amer. Math. Soc. 359 (2007) 6127–6142] on the analog of the Erdös–Falconer distance problem in the case of a finite field of characteristic [Formula: see text], where [Formula: see text] is an odd prime, without using estimates for Kloosterman sums. We also address the case of characteristic 2.

ON SUM OF PRODUCTS AND THE ERDŐS DISTANCE PROBLEM OVER FINITE FIELDS

For a prime powerq, let 𝔽qbe the finite field ofqelements. We show that 𝔽*q⊆d𝒜2for almost every subset 𝒜⊂𝔽qof cardinality ∣𝒜∣≫q1/d. Furthermore, ifq=pis a prime, and 𝒜⊆𝔽pof cardinality ∣𝒜∣≫p1/2(logp)1/d, thend𝒜2contains both large and small residues. We also obtain some results of this type for the Erdős distance problem over finite fields.

Explicit Ramsey Graphs and Erdős Distance Problems over Finite Euclidean and Non-Euclidean Spaces

We study the Erdős distance problem over finite Euclidean and non-Euclidean spaces. Our main tools are graphs associated to finite Euclidean and non-Euclidean spaces that are considered in Bannai-Shimabukuro-Tanaka (2004, 2007). These graphs are shown to be asymptotically Ramanujan graphs. The advantage of using these graphs is twofold. First, we can derive new lower bounds on the Erdős distance problems with explicit constants. Second, we can construct many explicit tough Ramsey graphs $R(3,k)$.

On some generalisations of the Erdős distance problem over finite fields

We use exponential sums to obtain new lower bounds on the number of distinct distances defined by all pairs of points (a, b) ∈ A × B for two given sets where is a finite field of q elements and n ≥ 1 is an integer.