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

TL;DR

This paper proves that sets of positive upper density in the Heisenberg group contain points at all sufficiently large Koranyi distances, extending Euclidean large distance results to a non-commutative setting.

Contribution

It establishes a Heisenberg group analogue of Bourgain's Euclidean large distance set theorem and analyzes spectral decay of surface measure coefficients in high frequency regimes.

Findings

Sets of positive upper density realize all large Koranyi distances.

Decay estimates for spectral coefficients in high frequency are obtained.

Positive upper density cannot be improved further.

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.