scholarly journals Erdös distance problem in vector spaces over finite fields

2007 ◽  
Vol 359 (12) ◽  
pp. 6127-6143 ◽  
Author(s):  
A. Iosevich ◽  
M. Rudnev
Keyword(s):  
Finite Fields ◽  
Vector Spaces ◽  
Distance Problem ◽  
Erdős Distance Problem ◽  
Erdos Distance Problem

The analog of the Erdös distance problem in finite fields

2017 ◽  
Vol 13 (09) ◽  
pp. 2319-2333
Author(s):  
S. D. Adhikari ◽  
Anirban Mukhopadhyay ◽  
M. Ram Murty
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Kloosterman Sums ◽  
Vector Spaces ◽  
Characteristic 2 ◽  
Distance Problem ◽  
Erdős Distance Problem ◽  
Erdos Distance Problem

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.

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

2011 ◽  
Vol 84 (1) ◽  
pp. 1-9
Author(s):  
LE ANH VINH
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Prime Power ◽  
Sum Of Products ◽  
Distance Problem ◽  
Erdős Distance Problem ◽  
Erdos Distance Problem

AbstractFor 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.

scholarly journals On some generalisations of the Erdős distance problem over finite fields

2006 ◽  
Vol 73 (2) ◽  
pp. 285-292 ◽  
Author(s):  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Exponential Sums ◽  
Distance Problem ◽  
Erdős Distance Problem ◽  
Erdos Distance Problem

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.

Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields

2012 ◽  
Vol 64 (5) ◽  
pp. 1036-1057 ◽  
Author(s):  
Doowon Koh ◽  
Chun-Yen Shen
Keyword(s):  
Finite Fields ◽  
Three Dimensional ◽  
Vector Spaces ◽  
Extension Problem ◽  
Dimensional Vector ◽  
Complete Mapping ◽  
Plane Passing ◽  
Size Condition ◽  
Distance Problem ◽  
Homogeneous Varieties

Abstract In this paper we study the extension problem, the averaging problem, and the generalized Erdős-Falconer distance problem associated with arbitrary homogeneous varieties in three dimensional vector spaces over finite fields. In the case when the varieties do not contain any plane passing through the origin, we obtain the best possible results on the aforementioned three problems. In particular, our result on the extension problem modestly generalizes the result by Mockenhaupt and Tao who studied the particular conical extension problem. In addition, investigating the Fourier decay on homogeneous varieties enables us to give complete mapping properties of averaging operators. Moreover, we improve the size condition on a set such that the cardinality of its distance set is nontrivial.

The Erdős Distance Problem

2010 ◽  
Author(s):  
Julia Garibaldi ◽  
Alex Iosevich ◽  
Steven Senger
Keyword(s):  
Distance Problem ◽  
Erdős Distance Problem ◽  
Erdos Distance Problem
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