2025/07/20 by Raani, K S Senthil, Singh, Rajesh K.
#26A33 (Primary) #42B20 #43A85 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.14917
We prove that any measurable set in the Heisenberg group, ℍn, 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, \cite1986Bourgain. Along the way, to prove our main theorem, we give the ``decay" of the coefficients Rk(λ, σ), appearing in the spectral decomposition of the group Fourier transform, σ(λ) = ∑k=0∞ Rk(λ, σ) Pk(λ), of the surface measure σ on the Korányi sphere in ℍn, in a certain ``high frequency" region, that is, when 2(2k+n) |λ| ≫ 1; which seems to be new in the literature. We also show that the positive upper density cannot be qualitatively improved further.