Number Theory

    

On the Erdh{o}s Distance Problem

Authors: Theophilus Agama

In this paper, using the method of compression, we recover the lower bound for the ErdH{o}s unit distance problem and provide an alternative proof to the distinct distance conjecture. In particular, in $mathbb{R}^k$ for all $kgeq 2$, we have begin{align}# bigg{||vec{x_j}-vec{x_t}||:~||vec{x_j}-vec{x_t}||=1,~1leq t,j leq n,~vec{x_j},~vec{x}_t in mathbb{R}^kbigg}gg_k frac{sqrt{k}}{2}n^{1+o(1)}.onumberend{align}We also show thatbegin{align}# bigg{d_j:d_j=||vec{x_s}-vec{y_t}||,~d_jeq d_i,~1leq s,tleq nbigg}gg_k frac{sqrt{k}}{2}n^{frac{2}{k}-o(1)}.onumber end{align}These lower bounds generalizes the lower bounds of the ErdH{o}s unit distance and the distinct distance problem to higher dimensions.

Comments: 8 Pages. This paper has been technically and substantially improved.

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Submission history

[v1] 2020-03-01 04:52:23
[v2] 2021-08-07 19:07:12
[v3] 2024-03-19 05:59:00

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