# Distances in sets of positive Kor\\'anyi upper density in Heisenberg Group

Source: [Distances in sets of positive Kor\\'anyi upper density in Heisenberg Group](<https://www.sota2.com/research/paper/68fa5dea1e209131ce9ddc76>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T18:29:04.534Z

Authors: K S Senthil Raani, Rajesh K. Singh

Published: 2025-07-20T11:15:03.000Z

## Abstract

We prove that any measurable set in the Heisenberg group, $\\mathbb{H}^n$, of positive upper density has the property that all sufficiently large real numbers are realised as the Kor\\'anyi distance between points in that set. The result can be seen as a Heisenberg group analogue to a corresponding Euclidean large distance set result in the $1986$ paper of Bourgain, \\cite{1986Bourgain}.   Along the way, to prove our main theorem, we give the \`\`decay" of the coefficients $R\_{k}(\\lambda, \\sigma)$, appearing in the spectral decomposition of the group Fourier transform, $\\hat{\\sigma}(\\lambda)$ $= \\sum\_{k=0}^{\\infty} R\_{k}(\\lambda, \\sigma) \\mathcal{P}\_{k}(\\lambda)$, of the surface measure $\\sigma$ on the Kor\\'anyi sphere in $\\mathbb{H}^n$, in a certain \`\`high frequency" region, that is, when $2(2k+n) \|\\lambda\| \\gg 1$; which seems to be new in the literature. We also show that the positive upper density cannot be qualitatively improved further.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2507.14917>)

[PDF](<https://arxiv.org/pdf/2507.14917.pdf>)