Erdös--Falconer Distance Problem under Hamming Metric in Vector Spaces over Finite Fields

2020 ◽  
Vol 34 (4) ◽  
pp. 2208-2220
Author(s):  
Zixiang Xu ◽  
Gennian Ge
Keyword(s):  
Finite Fields ◽  
Vector Spaces ◽  
Hamming Metric ◽  
Distance Problem

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 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 A note on extensions of multilinear maps defined on multilinear varieties

2021 ◽  
pp. 1-26
Author(s):  
W. T. Gowers ◽  
L. Milićević
Keyword(s):  
Finite Fields ◽  
Vector Spaces ◽  
Restriction Map ◽  
Zero Set ◽  
Dimensional Vector ◽  
Prime Field ◽  
Finite Dimensional ◽  
Multilinear Maps ◽  
Multilinear Map

Abstract Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$ . A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$ , $i\not =c$ , the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$ . Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

scholarly journals Restriction Operators Acting on Radial Functions on Vector Spaces over Finite Fields

2014 ◽  
Vol 57 (4) ◽  
pp. 834-844
Author(s):  
Doowon Koh
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Algebraic Varieties ◽  
Test Functions ◽  
Radial Functions ◽  
Zero Radius ◽  
Intersection Points ◽  
Field Setting

AbstractWe study Lp → Lr restriction estimates for algebraic varieties V in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties V lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, assuming that the varieties V are defined in even dimensional spaces and have few intersection points with the sphere of zero radius, we also obtain the conjectured exponents for all radial test functions.

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