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SUMMARY:Christian Tafula (Universit\\'e de Montr\\'eal)
DTSTART:20250521T160000Z
DTEND:20250521T162500Z
DTSTAMP:20260423T010356Z
UID:CANT2025/31
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/31/
 ">Waring--Goldbach subbases with prescribed representation functions</a>\n
 by Christian Tafula (Universit\\'e de Montr\\'eal) as part of Combinatoria
 l and additive number theory (CANT 2025)\n\nLecture held in CUNY Graduate 
 Center - Science Center (4th floor).\n\nAbstract\nWe investigate represent
 ation functions $r_{A\,h}(n)$ of subsets $A$ of \\( k \\)-th powers \\( \\
 mathbb{N}^k \\) and \\( k \\)-th powers of primes \\( \\mathbb{P}^k \\). B
 uilding on work of Vu\, Wooley\, and others\, we prove that for \\( h \\ge
 q h_k = O(8^k k^2) \\) and regularly varying \\( F(n) \\) satisfying \\( \
 \lim_{n\\to\\infty} F(n)/\\log n = \\infty \\)\, there exists \\( A \\subs
 eteq \\mathbb{N}^k \\) such that\n \\[ r_{A\,h}(n) \\sim \\mathfrak{S}_{k\
 ,h}(n) F(n)\, \\]\n where $\\mathfrak{S}_{k\,h}(n)$ is the singular series
  associated to Waring's problem. In the case of prime powers\, we obtain a
 nalogous results for \\( F(n) = n^{\\kappa} \\). For \\( F(n) = \\log n \\
 )\, we prove that for every \\( h \\geq 2k^2(2\\log k + \\log\\log k + O(1
 )) \\)\, there exists \\( A \\subseteq \\mathbb{P}^k \\) such that \\( r_{
 A\,h}(n) \\asymp \\log n \\)\, showing the existence of thin subbases of p
 rime powers.\n
LOCATION:https://researchseminars.org/talk/CANT2025/31/
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