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Zane Kun Li

Publications and source records attributed to Zane Kun Li.

15 recordsLinked to original sources
arXiv2025

A weighted formulation of refined decoupling and inequalities of Mizohata-Takeuchi-type for the moment curve

Let $Γ$ be a compact patch of a well-curved $C^{n+1}$ curve in $\mathbb{R}^n$ with induced Lebesgue measure ${\rm d} λ$, and let $g \mapsto \widehat{g \,{\rm d}λ}$ be the Fourier extension operator for $Γ$. Then we have, for arbitrary non-negative weights $w$, \begin{equation*} \int_{B_R} |\widehat{g \,{\rm d}λ}|^2w \leq C_{n,a} R^{a} \sup_S \left(\int_S w\right)\int_Γ|g|^2 \, {\rm d} λ \end{equation*} for any $a> \frac{n-3}{2} + \frac{2}{n} - \frac{2}{n^2(n+1)}$, where the $\sup$ is over all $1$-neighbourhoods $S$ of hyperplanes whose normals are parallel to the tangent at some point of $Γ$. This represents partial progress on the Mizohata-Takeuchi conjecture for curves in dimensions $n \geq 3$, improving upon the exponent $a=n-1$ which can be obtained as a consequence of the Agmon-Hörmander trace inequality. Our main tool in establishing this inequality will be a weighted formulation of refined decoupling for well-curved curves. We also discuss the sharpness of the exponents we obtain in this and in auxiliary results, and further explore this in the context of axiomatic decoupling for curves.

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arXiv2025

An improved example for an autoconvolution inequality

We give a nonnegative step function with 575 equally spaced intervals such that $$\frac{\|f \ast f\|_{L^{2}(\mathbb{R})}^{2}}{\|f \ast f\|_{L^{\infty}(\mathbb{R})}\|f \ast f\|_{L^{1}(\mathbb{R})}} \geq 0.901564.$$ This improves upon a recent result of Deepmind's AlphaEvolve, which found a nonnegative step function with 50 equally space intervals for which the left hand side is $\geq 0.8962$. Our function was found using simulated annealing and gradient based methods rather than using large language models.

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arXiv2024

Strichartz inequalities: some recent developments

Strichartz inequalities, originating from Fourier restriction theory, play a central role in the analysis of dispersive partial differential equations. They serve as a cornerstone for many subsequent developments. We survey some of them in memory of Strichartz, highlighting connections to recent developments in Fourier decoupling.

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arXiv2023

A short proof of $\ell^2$ decoupling for the moment curve

We give a short and elementary proof of the $\ell^{2}$ decoupling inequality for the moment curve in $\mathbb{R}^k$, using a bilinear approach inspired by the nested efficient congruencing argument of Wooley (arXiv:1708.01220).

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arXiv2023

An introduction to decoupling and harmonic analysis over $\mathbb{Q}_p$

The goal of this expository paper is to provide an introduction to decoupling by working in the simpler setting of decoupling for the parabola over $\mathbb{Q}_p$. Over $\mathbb{Q}_p$, commonly used heuristics in decoupling are significantly easier to make rigorous over $\mathbb{Q}_p$ than over $\mathbb{R}$ and such decoupling theorems over $\mathbb{Q}_p$ are still strong enough to derive interesting number theoretic conclusions.

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arXiv2021

Decoupling for fractal subsets of the parabola

We consider decoupling for a fractal subset of the parabola. We reduce studying $l^{2}L^{p}$ decoupling for a fractal subset on the parabola $\{(t, t^2) : 0 \leq t \leq 1\}$ to studying $l^{2}L^{p/3}$ decoupling for the projection of this subset to the interval $[0, 1]$. This generalizes the decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the sparsity and fractal like structure, this allows us to improve upon Bourgain-Demeter's decoupling theorem for the parabola. In the case when $p/3$ is an even integer we derive theoretical and computational tools to explicitly compute the associated decoupling constant for this projection to $[0, 1]$. Our ideas are inspired by the recent work on ellipsephic sets by Biggs using nested efficient congruencing.

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arXiv2021

Improved discrete restriction for the parabola

Using ideas from Guth-Maldague-Wang and working over $\mathbb{Q}_p$, we show that the discrete restriction constant for the parabola is $O_{\varepsilon}((\log M)^{2 + \varepsilon})$.

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arXiv2020

An $l^2$ decoupling interpretation of efficient congruencing: the parabola

We give a new proof of $l^2$ decoupling for the parabola inspired from efficient congruencing. Making quantitative this proof matches a bound obtained by Bourgain for the discrete restriction problem for the parabola. We illustrate similarities and differences between this new proof and efficient congruencing and the proof of decoupling by Bourgain and Demeter. We also show where tools from decoupling such as $l^2 L^2$ decoupling, Bernstein, and ball inflation come into play.

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arXiv2020

Effective $l^2$ decoupling for the parabola

We make effective $l^2 L^p$ decoupling for the parabola in the range $4 < p < 6$. In an appendix joint with Jean Bourgain, we apply the main theorem to prove the conjectural bound for the sixth-order correlation of the integer solutions of the equation $x^2 + y^2 = m$ in an extremal case. This proves unconditionally a result that was proven by Bombieri and Bourgain under the hypotheses of the Birch and Swinnerton-Dyer conjecture and the Riemann Hypothesis for $L$-functions of elliptic curves over $\mathbb{Q}$.

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arXiv2013

Quadratic Twists of Elliptic Curves with 3-Selmer Rank 1

A weaker form of a 1979 conjecture of Goldfeld states that for every elliptic curve $E/\mathbb{Q}$, a positive proportion of its quadratic twists $E^{(d)}$ have rank 1. Using tools from Galois cohomology, we give criteria on E and d which force a positive proportion of the quadratic twists of E to have 3-Selmer rank 1 and global root number -1. We then give four nonisomorphic infinite families of elliptic curves $E_{m, n}$ which satisfy these criteria. Conditional on the rank part of the Birch and Swinnerton-Dyer conjecture, this verifies the aforementioned conjecture for infinitely many elliptic curves. Our elliptic curves are easy to give explicitly and we state precisely which quadratic twists d to use. Furthermore, our methods have the potential of being generalized to elliptic curves over other number fields.

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arXiv2013

Arithmetic Properties of Picard-Fuchs Equations and Holonomic Recurrences

The coefficient series of the holomorphic Picard-Fuchs differential equation associated with the periods of elliptic curves often have surprising number-theoretic properties. These have been widely studied in the case of the torsion-free, genus zero congruence subgroups of index 6 and 12 (e.g. the Beauville families). Here, we consider arithmetic properties of the Picard-Fuchs solutions associated to general elliptic families, with a particular focus on the index 24 congruence subgroups. We prove that elliptic families with rational parameters admit linear reparametrizations such that their associated Picard-Fuchs solutions lie in Z[[t]]. A sufficient condition is given such that the same holds for holomorphic solutions at infinity. An Atkin-Swinnerton-Dyer congruence is proven for the coefficient series attached to Γ_1(7). We conclude with a consideration of asymptotics, wherein it is proved that many coefficient series satisfy asymptotic expressions of the form u_n \sim \ell λ^n/n. Certain arithmetic results extend to the study of general holonomic recurrences.

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arXiv2013

Elliptic Curves with Full 2-Torsion and Maximal Adelic Galois Representations

In 1972, Serre showed that the adelic Galois representation associated to a non-CM elliptic curve over a number field has open image in GL_2(\hat{Z}). In Greicius' thesis, he develops necessary and sufficient criteria for determining when this representation is actually surjective and exhibits such an example. However, verifying these criteria turns out to be difficult in practice; Greicius describes tests for them that apply only to semistable elliptic curves over a specific class of cubic number fields. In this paper, we extend Greicius' methods in several directions. First, we consider the analogous problem for elliptic curves with full 2-torsion. Following Greicius, we obtain necessary and sufficient conditions for the associated adelic representation to be maximal and also develop a battery of computationally effective tests that can be used to verify these conditions. We are able to use our tests to construct an infinite family of curves over Q(alpha) with maximal image, where alpha is the real root of x^3 + x + 1. Next, we extend Greicius' tests to more general settings, such as non-semistable elliptic curves over arbitrary cubic number fields. Finally, we give a general discussion concerning such problems for arbitrary torsion subgroups.

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