Distances in sets of positive Korányi upper density in Heisenberg Group
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913950121066496 |
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| author | Raani, K S Senthil Singh, Rajesh K. |
| author_facet | Raani, K S Senthil Singh, Rajesh K. |
| contents | 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 $R_{k}(λ, σ)$, appearing in the spectral decomposition of the group Fourier transform, $\hatσ(λ)$ $= \sum_{k=0}^{\infty} R_{k}(λ, σ) \mathcal{P}_{k}(λ)$, of the surface measure $σ$ on the Korányi sphere in $\mathbb{H}^n$, in a certain ``high frequency" region, that is, when $2(2k+n) |λ| \gg 1$; which seems to be new in the literature. We also show that the positive upper density cannot be qualitatively improved further. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14917 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distances in sets of positive Korányi upper density in Heisenberg Group Raani, K S Senthil Singh, Rajesh K. Classical Analysis and ODEs 42B20, 26A33 (Primary), 43A85 (Secondary) 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 $R_{k}(λ, σ)$, appearing in the spectral decomposition of the group Fourier transform, $\hatσ(λ)$ $= \sum_{k=0}^{\infty} R_{k}(λ, σ) \mathcal{P}_{k}(λ)$, of the surface measure $σ$ on the Korányi sphere in $\mathbb{H}^n$, in a certain ``high frequency" region, that is, when $2(2k+n) |λ| \gg 1$; which seems to be new in the literature. We also show that the positive upper density cannot be qualitatively improved further. |
| title | Distances in sets of positive Korányi upper density in Heisenberg Group |
| topic | Classical Analysis and ODEs 42B20, 26A33 (Primary), 43A85 (Secondary) |
| url | https://arxiv.org/abs/2507.14917 |