Distances from points to planes
P. Birklbauer, A. Iosevich, T. Pham

TL;DR
This paper establishes a lower bound on the number of distances from points to affine hyperplanes in finite fields, improving previous bounds in higher dimensions.
Contribution
It proves a new lower bound on the number of point-plane distances in finite fields, enhancing earlier results for dimensions three and above.
Findings
Lower bound of q/2 on distances when |E||F| > q^{d+1}
Significant improvement over previous exponents in dimensions ≥ 3
Applicable to finite field geometries with affine hyperplanes
Abstract
We prove that if , , , the set of affine -dimensional planes in , then if , where the set of distances from points in to lines in . In dimension three and higher this significantly improves the exponent obtained by Pham, Phuong, Sang, Vinh and Valculescu.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Finite Group Theory Research · Coding theory and cryptography
