communication: The Gauss Circle Problem for periodic measures
speaker: Giacomo Del Nin (Università di Pisa)abstract: The classic Gauss Circle Problem asks to determine the number $N(r)$ of points in the planar integer lattice that are contained inside the disk $Br$ of radius $r$ centered at the origin. While at the leading order (as $r$ goes to infinity) $N(r)$ goes like the area of the disk, the estimate of the remainder term $N(r)-\pi r2$ is a subtler issue.
A proof by C.S. Herz (1962) uses the Fourier transform to give the non-optimal estimate $O(r{23})$, but has the advantage of being rather flexible: we will show that the same estimate holds for any measure $\mu$ which is periodic with respect to the integer lattice, that is we have $\mu(Br)=cr2+O(r{23})$ for a suitable constant $c$.
An analogous result holds in any dimension of the ambient space.
timetable:
Thu 6 Sep, 17:01 - 17:40, Sala Riunioni del Centro De Giorgi
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