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SUMMARY:Bartosz Langowski (IU Bloomington)
DTSTART:20201019T210000Z
DTEND:20201019T220000Z
DTSTAMP:20260423T005742Z
UID:OARS/3
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OARS/3/">Lat
 tice point problems\, equidistribution and ergodic theorems for certain ar
 ithmetic spheres</a>\nby Bartosz Langowski (IU Bloomington) as part of OAR
 S Online Analysis Research Seminar\n\n\nAbstract\nLet $\\lambda\\in\\Z_+$ 
 be a positive integer and define the set\n$\\mathbf S_{2}^3(\\lambda)$ of 
 all lattice points on a two-dimensional\nsphere with radius $\\lambda^{1/2
 }$ by\n\\[\n\\mathbf S_{2}^3(\\lambda)\n:=\n\\{x \\in \\mathbb Z^3 : x_1^2
  +x_2^2 +x_3^2 = \\lambda \\}.\n\\]\nThe study of the behavior of $\\mathb
 f S_{2}^3(\\lambda)$ as\n$\\lambda\\to\\infty$ is a central problem in num
 ber theory\, which has\ngone through a period of considerable change and d
 evelopment in the\nlast three decades.\n\n\nIn the recent work with A. Ios
 evich\, M. Mirek and T.Z. Szarek we  consider perturbations of\nthe discre
 te spheres $\\mathbf S_2^3(\\lambda)$. In particular\, for $c\\in (1\,2)$\
 nwe derive  an asymptotic formula for the number of lattice points in the 
 sets\n\\[\n\\mathbf S_{c}^3(\\lambda)\n:=\n\\{x \\in \\mathbb Z^3 : \\lflo
 or |x_1|^c \\rfloor + \\lfloor |x_2|^c \\rfloor + \\lfloor |x_3|^c \\rfloo
 r= \\lambda \\}\n\\quad \\text{with}\\quad \\lambda\\in\\mathbb Z_+\;\n\\]
 \nwhich can be thought of as a perturbation of the classical Waring proble
 m in three variables. Then we  use the obtained asymptotic formula to stud
 y  norm and\npointwise convergence  of  the ergodic averages\n\\[\n\\frac{
 1}{\\#\\mathbf S_{c}^3(\\lambda)}\\sum_{n\\in \\mathbf S_{c}^3(\\lambda)}f
 (T_1^{n_1}T_2^{n_2}T_3^{n_3}x)\n\\quad \\text{as}\\quad \\lambda\\to\\inft
 y\;\n\\]\nwhere $T_1\, T_2\, T_3:X\\to X$ are commuting invertible and\nme
 asure-preserving transformations of a $\\sigma$-finite measure space\n$(X\
 , \\nu)$ for any function  $f\\in L^p(X)$ with $p>\\frac{11-4c}{11-7c}$. F
 inally\, we study the equidistribution problem corresponding to the\nspher
 es $\\mathbf S_{c}^3(\\lambda)$.\n
LOCATION:https://researchseminars.org/talk/OARS/3/
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