arXiv subjects

W. Burstein

Publications and source records attributed to W. Burstein.

3 recordsLinked to original sources

arXiv2026

Elliptic curves, Fourier ratio, and sampling complexity

We study the normalized Frobenius trace associated with the Legendre family of elliptic curves over $\mathbb F_p$ from the point of view of Fourier complexity. If \[ f(t)=\frac{a_p(E_t)}{\sqrt p}, \qquad E_t:\ y^2=x(x-1)(x-t), \] with $f(0)=f(1)=0$, then \[ \frac{\|\widehat f\|_1}{\|\widehat f\|_2}\asymp \sqrt p. \] More precisely, the Fourier transform of $f$ has squared $\ell^2$ norm of order $p$ while its individual coefficients remain uniformly bounded. It follows that no Fourier model supported on fewer than a sufficiently small constant multiple of $p$ frequencies can approximate $f$ in $\ell^2$ with error smaller than a fixed proportion of $\|f\|_2$. We also show that the Fourier magnitude profile of $f$ supports a family of at least $\exp(cp)$ real-valued functions with identical Fourier magnitudes and identical Fourier ratio, any two of which are separated by at least $c\sqrt p$ in $\ell^2$. Consequently, every deterministic reconstruction procedure that recovers all members of this family from bounded-precision point evaluations must use at least $c_Bp$ samples, where $c_B>0$ depends only on the number of bits used to encode each observation. The arithmetic input is unconditional and relies only on the Weil bound for mixed character sums, the evaluation of the quadratic Gauss sum, and elementary character identities.

math.NT↗

arXiv2026

Arithmetic functions and learning theory

We establish a connection between analytic number theory and computational learning theory by showing that the M\"obius function belongs to a class of functions that is statistically hard to learn from random samples. Let $\mu_R$ denote the restriction of the M\"obius function to the squarefree integers in $\{1,\dots,R\}$. Using a recent lower bound of Pandey and Radziwi{\l}{\l} for the $L^1$ norm of exponential sums with M\"obius coefficients, we prove that \[ \FR(\mu_R) \gg R^{-1/4-\epsilon} \] for every $\epsilon>0$. We then show that, for a suitable absolute constant $c_0>0$, the class of $\{-1,1\}$-valued functions on the squarefree integers with Fourier Ratio at least $c_0$ has Vapnik--Chervonenkis dimension at least $cR$. It follows that any distribution-independent learning algorithm that succeeds uniformly on the class $\mathcal{H}_R(\eta_R)$ containing $\mu_R$, where $\eta_R \to 0$, requires at least $\Omega(R)$ samples. We also discuss a conditional improvement under a strong uniform bound for additive twists of the M\"obius function, and we note that the same method applies to the Liouville function.

math.NT↗

arXiv2025

The Fourier Ratio and complexity of signals

We study the Fourier ratio of a signal $f:\mathbb Z_N\to\mathbb C$, \[ \mathrm{FR}(f)\ :=\ \sqrt{N}\,\frac{\|\widehat f\|_{L^1(\mu)}}{\|\widehat f\|_{L^2(\mu)}} \ =\ \frac{\|\widehat f\|_1}{\|\widehat f\|_2}, \] as a simple scalar parameter governing Fourier-side complexity, structure, and learnability. Using the Bourgain--Talagrand theory of random subsets of orthonormal systems, we show that signals concentrated on generic sparse sets necessarily have large Fourier ratio, while small $\mathrm{FR}(f)$ forces $f$ to be well-approximated in both $L^2$ and $L^\infty$ by low-degree trigonometric polynomials. Quantitatively, the class $\{f:\mathrm{FR}(f)\le r\}$ admits degree $O(r^2)$ $L^2$-approximants, which we use to prove that small Fourier ratio implies small algorithmic rate--distortion, a stable refinement of Kolmogorov complexity.

math.CA↗