Horizontal Kakeya maximal operators in finite Heisenberg groups: Exact exponents and applications
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918366788190208 |
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| author | Pham, Thang Pinamonti, Andrea Tran, Dung The Xue, Boqing |
| author_facet | Pham, Thang Pinamonti, Andrea Tran, Dung The Xue, Boqing |
| contents | We study Kakeya maximal operators associated with horizontal lines in finite Heisenberg groups $\mathbb H_n(\mathbb F_q)$. For the operator parameterized only by projective horizontal directions, we show that projection to $\mathbb F_q^{2n}$ reduces the problem to the affine finite field Kakeya maximal operator, and we determine the exact $\ell^u \to \ell^v$ growth exponent for all $n$ and all $1 \le u,v \le \infty$. We then introduce a refined-direction operator that also records the central slope of a horizontal line. In $\mathbb H_1(\mathbb F_q)$, we prove the sharp $\ell^2 \to \ell^2$ estimate \[ \|M_{\mathbb H_1}^{\mathrm{rd}}F\|_{\ell^2(D_1)} \lesssim q^{1/2}\|F\|_{\ell^2(\mathbb H_1(\mathbb F_q))}, \] deduce the exact mixed-norm exponent formula, and obtain lower bounds for horizontal Heisenberg Kakeya sets with prescribed refined directions. The argument is purely Fourier-analytic and does not use the polynomial method. An outlook toward a new approach to the affine Kakeya problem in $\mathbb{F}_q^3$ will be discussed in this paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_02111 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Horizontal Kakeya maximal operators in finite Heisenberg groups: Exact exponents and applications Pham, Thang Pinamonti, Andrea Tran, Dung The Xue, Boqing Combinatorics Classical Analysis and ODEs Group Theory Number Theory We study Kakeya maximal operators associated with horizontal lines in finite Heisenberg groups $\mathbb H_n(\mathbb F_q)$. For the operator parameterized only by projective horizontal directions, we show that projection to $\mathbb F_q^{2n}$ reduces the problem to the affine finite field Kakeya maximal operator, and we determine the exact $\ell^u \to \ell^v$ growth exponent for all $n$ and all $1 \le u,v \le \infty$. We then introduce a refined-direction operator that also records the central slope of a horizontal line. In $\mathbb H_1(\mathbb F_q)$, we prove the sharp $\ell^2 \to \ell^2$ estimate \[ \|M_{\mathbb H_1}^{\mathrm{rd}}F\|_{\ell^2(D_1)} \lesssim q^{1/2}\|F\|_{\ell^2(\mathbb H_1(\mathbb F_q))}, \] deduce the exact mixed-norm exponent formula, and obtain lower bounds for horizontal Heisenberg Kakeya sets with prescribed refined directions. The argument is purely Fourier-analytic and does not use the polynomial method. An outlook toward a new approach to the affine Kakeya problem in $\mathbb{F}_q^3$ will be discussed in this paper. |
| title | Horizontal Kakeya maximal operators in finite Heisenberg groups: Exact exponents and applications |
| topic | Combinatorics Classical Analysis and ODEs Group Theory Number Theory |
| url | https://arxiv.org/abs/2603.02111 |