Author

Xi Cen

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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