4.5 Article

Group action and L2-norm estimates of geometric problems

Related references

Note: Only part of the references are listed.
Article Mathematics

On the pinned distances problem in positive characteristic

Brendan Murphy et al.

Summary: The Erdos-Falconer distance problem for a set A in F2 is studied, where F is a field with positive characteristic p. If F=Fp and the cardinality of A exceeds p5/4, it is proven that A determines an asymptotically full proportion of the feasible p distances. For small sets A, namely when |A|<= p4/3 over any F, it is proven that either A determines >>|A|2/3 distances, or A lies on an isotropic line. The results proved are for pinned distances in both large and small sets.

JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES (2022)

Article Mathematics

On Restriction Estimates for the Zero Radius Sphere over Finite Fields

Alex Iosevich et al.

Summary: This paper completely solves the L-2 -> L-r extension conjecture for the zero radius sphere over finite fields and obtains the sharp L-p -> L-4 extension estimate for non-zero radii spheres over finite fields, significantly improving the previous results.

CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES (2021)

Article Mathematics

Existence of similar point configurations in thin subsets of Rd

Allan Greenleaf et al.

Summary: The paper investigates the existence of similar and multi-similar point configurations in sets of fractional Hausdorff dimension in Euclidean space. The authors prove the existence of many pairs of (k + 1)-point configurations which are similar by the scaling factor r, and even show the existence of multi-similar configurations of any multiplicity. These results offer new insights and extensions to theorems for sets of positive density in R-d.

MATHEMATISCHE ZEITSCHRIFT (2021)

Article Mathematics

A sharp exponent on sum of distance sets over finite fields

Doowon Koh et al.

Summary: We study a variant of the Erdos-Falconer distance problem in finite fields and provide results on the covering properties of distance sets, including improvements in odd dimensional spaces and on spheres. Additionally, we present a weak version of the Erdos-Falconer distance conjecture in four-dimensional vector spaces. The novelty of our method lies in the connection with additive energy bounds of sets on spheres or paraboloids.

MATHEMATISCHE ZEITSCHRIFT (2021)

Article Mathematics

Extension theorems and a connection to the Erdos-Falconer distance problem over finite fields

Doowon Koh et al.

Summary: The paper introduces new finite field extension theorems for paraboloids and spheres, showing a different extension phenomenon between spheres and paraboloids in odd dimensions. Additionally, it explores the connection between the restriction conjecture associated to paraboloids and the Erdos-Falconer distance conjecture over finite fields. Finally, it proves the Erdos-Falconer distance conjecture in odd dimensional spaces when studying distances between sets on varieties and arbitrary sets in vector spaces over finite fields.

JOURNAL OF FUNCTIONAL ANALYSIS (2021)

Article Mathematics

An improved result for Falconer's distance set problem in even dimensions

Xiumin Du et al.

Summary: It is shown that the distance set of a compact set E in R-d with Hausdorff dimension larger than d/2 + 1/4 has positive Lebesgue measure. This result represents an improvement towards Falconer's distance set conjecture in even dimensions.

MATHEMATISCHE ANNALEN (2021)

Article Mathematics, Applied

An Asymmetric Bound for Sum of Distance Sets

Daewoong Cheong et al.

Summary: The study analyzes the relationship between distance sets under certain conditions using additive energies of sets, and provides an improved result in the unbalanced case, which is essential sharp in odd dimensions. The optimal L-2 restriction theorem for the sphere of zero radius is a key tool in the proofs.

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS (2021)

Article Mathematics

On Falconer's distance set problem in the plane

Larry Guth et al.

INVENTIONES MATHEMATICAE (2020)

Article Mathematics

An L2-identity and pinned distance problem

Bochen Liu

GEOMETRIC AND FUNCTIONAL ANALYSIS (2019)

Article Mathematics, Applied

GROUP ACTIONS, THE MATTILA INTEGRAL AND APPLICATIONS

Bochen Liu

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY (2019)

Article Mathematics, Applied

NEW RESULTS ON SUM-PRODUCT TYPE GROWTH OVER FIELDS

Brendan Murphy et al.

MATHEMATIKA (2019)

Article Mathematics, Applied

Group actions and geometric combinatorics in IFqd

Michael Bennett et al.

FORUM MATHEMATICUM (2017)

Article Mathematics, Applied

SIMPLICES OVER FINITE FIELDS

Hans Parshall

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY (2017)

Article Mathematics, Applied

On the additive energy of the distance set in finite fields

Igor E. Shparlinski

FINITE FIELDS AND THEIR APPLICATIONS (2016)

Article Mathematics

A group-theoretic viewpoint on Erdos-Falconer problems and the Mattila integral

Allan Greenleaf et al.

REVISTA MATEMATICA IBEROAMERICANA (2015)

Article Mathematics

Distance sets of two subsets of vector spaces over finite fields

Doowon Koh et al.

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY (2014)

Article Mathematics, Applied

Another sum-product estimate in finite fields

A. Balog

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS (2013)

Article Mathematics

Pinned distance sets, k-simplices, Wolff's exponent in finite fields and sum-product estimates

Jeremy Chapman et al.

MATHEMATISCHE ZEITSCHRIFT (2012)

Article Mathematics

ON KALEIDOSCOPIC PSEUDO-RANDOMNESS OF FINITE EUCLIDEAN GRAPHS

Le Anh Vinh

DISCUSSIONES MATHEMATICAE GRAPH THEORY (2012)

Article Mathematics

Ubiquity of simplices in subsets of vector spaces over finite fields

Derrick Hart et al.

Analysis Mathematica (2008)