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SUMMARY:Peter Humphries (University College London)
DTSTART:20200528T120000Z
DTEND:20200528T130000Z
DTSTAMP:20260423T005809Z
UID:AGNTISTA/6
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/AGNTISTA/6/"
 >Small scale equidistribution of lattice points on the sphere</a>\nby Pete
 r Humphries (University College London) as part of Algebraic Geometry and 
 Number Theory seminar - ISTA\n\n\nAbstract\nConsider the projection onto t
 he unit sphere in $\\mathbb{R}^3$ of the set of lattice points $(x_1\, x_2
 \, x_3) \\in \\mathbb{Z}^3$ lying on the sphere of radius $\\sqrt{n}$. Duk
 e and Schulze-Pillot showed in 1990 that these points equidistribute on th
 e sphere as $n \\to \\infty$. We study a small scale refinement of this th
 eorem\, where one asks whether these points equidistribute in subsets of t
 he sphere whose surface area shrinks as $n$ grows. A particular case of th
 is is a conjecture of Linnik\, which states that for all $\\delta > 0$\, t
 he equation $x_1^2 + x_2^2 + x_3^2 = n$ has a solution with $|x_3| < n^{\\
 delta}$ for all sufficiently large $n$. We make nontrivial progress toward
 s this\, as well as proving an averaged form of this conjecture. This is j
 oint work with Maksym Radziwiłł.\n
LOCATION:https://researchseminars.org/talk/AGNTISTA/6/
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