Distances in sets of positive Korányi upper density in Heisenberg Group

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 Korányi 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 Korányi 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.