Author

Jun-Yong Zhang

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Preprint Oct 2026

A New Bound for the Gauss Circle Problem

Let $E(X)=\#\{(a,b)\in\Z^2:a^2+b^2\le X\}-\pi X$. We prove that $E(X)=O_\varepsilon(X^{209/670+\varepsilon})$ for every $\varepsilon>0$, uniformly for real $X\ge4$. This breaks the $5/16$ threshold suggested by Huxley as a possible limit of the Bombieri--Iwaniec method at the time. The proof introduces two refinements:...

Xi Cen, Xi-Tao Gao, Cheng Zhang et al. · 0 citations

Preprint Sep 2026

A sharp regularity threshold for Schr\"odinger maximal estimates on standard tori

We disprove almost everywhere convergence of the Schr\"odinger evolution on the standard torus $\mathbb{T}^d$ for initial data $f\in H^s(\mathbb{T}^d)$ when $s<d/(d+2)$, in all dimensions $d\ge2$. We construct normalized data with frequencies of size $N$ whose evolution attains size $N^{d/(d+2)}$ on a set of un...

Xi Cen, Xi-Tao Gao, Jun-Yong Zhang · 1 citation · ⚡1

Preprint Sep 2026

$L^p$ bounds for wave operators with critical electromagnetic potentials

We study the M\o ller wave operators for scaling critical electromagnetic Hamiltonians in the plane. For smooth transverse magnetic and angular electric potentials, with nonnegative angular operator and magnetic flux outside $\frac12\Z$, we prove that the wave operators relative to $-\Delta$ exist, are unitary on $L^2$...

Piero D'Ancona, Xi-Tao Gao, Jun-Yong Zhang · 1 citation