Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.05485 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913600628588544 |
|---|---|
| author | Edwin, Roni A. Lin, Allen |
| author_facet | Edwin, Roni A. Lin, Allen |
| contents | The Gauss circle problem asks for an approximation to the number of lattice points of $\mathbb{Z}^2$ contained in $B_r$, the disk of radius $r$ centered at the origin. Upper, lower, and average bounds have been established for this number-theoretic problem and have been generalized to any lattice in any dimension. We extend this problem to a more general class of structures known as Fourier quasicrystals. Recent work from Alon, Kummer, Kurasov, and Vinzant provides an upper bound $\#(Λ\cap B_r) = c_0\textrm{Vol}_d\left(B_r\right)+ O\left(r^{d-1}\right)$ for any Fourier quasicrystal $Λ\subset \mathbb{R}^d$ of density $c_0$, where $B_r$ is the $d$-dimensional ball of radius $r$. In this paper, we improve the upper bound for any uniformly discrete Fourier quasicrystal, by showing we can write $\#\left(Λ\cap B_r\right) = c_0\textrm{Vol}_d\left(B_r\right) + O\left(r^{θ(Λ)}\right)$, where $\frac{d-1}{2} < θ(Λ) < d-1$ is some exponent depending on $Λ$. In the special case $d = 2$, we also prove lower and upper bounds for the average of the error. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05485 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Gauss Circle Problem and Fourier Quasicrystals Edwin, Roni A. Lin, Allen Mathematical Physics The Gauss circle problem asks for an approximation to the number of lattice points of $\mathbb{Z}^2$ contained in $B_r$, the disk of radius $r$ centered at the origin. Upper, lower, and average bounds have been established for this number-theoretic problem and have been generalized to any lattice in any dimension. We extend this problem to a more general class of structures known as Fourier quasicrystals. Recent work from Alon, Kummer, Kurasov, and Vinzant provides an upper bound $\#(Λ\cap B_r) = c_0\textrm{Vol}_d\left(B_r\right)+ O\left(r^{d-1}\right)$ for any Fourier quasicrystal $Λ\subset \mathbb{R}^d$ of density $c_0$, where $B_r$ is the $d$-dimensional ball of radius $r$. In this paper, we improve the upper bound for any uniformly discrete Fourier quasicrystal, by showing we can write $\#\left(Λ\cap B_r\right) = c_0\textrm{Vol}_d\left(B_r\right) + O\left(r^{θ(Λ)}\right)$, where $\frac{d-1}{2} < θ(Λ) < d-1$ is some exponent depending on $Λ$. In the special case $d = 2$, we also prove lower and upper bounds for the average of the error. |
| title | The Gauss Circle Problem and Fourier Quasicrystals |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2412.05485 |