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SUMMARY:David Ross (University of Hawaii)
DTSTART:20250523T203000Z
DTEND:20250523T205500Z
DTSTAMP:20260423T024834Z
UID:CANT2025/55
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/55/
 ">Upper density and a theorem of Banach</a>\nby David Ross (University of 
 Hawaii) as part of Combinatorial and additive number theory (CANT 2025)\n\
 nLecture held in CUNY Graduate Center - Science Center (4th floor).\n\nAbs
 tract\nSuppose $A_n$ $(n\\in\\mathbb{N})$ is a sequence of sets in a finit
 ely-additive measure space which are uniformly bounded away from $0$\, $\\
 mu{A_n}\\ge a>0$ for all $n$.  Then there is a subsequence $A_{n_k}$\, whe
 re $\\{n_k\\}_k$ has upper Banach density $\\ge a$\, such that $\\mu\\bigc
 ap_{k<N}A_{n_k}\\ge a$ for every $N$.  Surprisingly\, this implies a densi
 ty-limit version of a representation theorem of Banach:\n\n\\textbf{Theore
 m:} Let $\\{\\\,f_n : n\\in\\mathbb{N}\\}$ be a uniformly\nbounded sequenc
 e of functions on a set $X$.  The following are equivalent:  (i)~$\\{f_n\\
 }_n$ weakly d-converges to $0$\; (ii)~for any sequence\n$\\{x_k : k\\in\\m
 athbb{N}\\}$ in $X$\, $d$-$\\!\\lim\\limits_{n\\to\\infty}\\liminf\\limits
 _{k\\to\\infty}f_n(x_k)=0$.\n\nHere ``d-" denotes a density limit.  Banach
 's non-density version of this theorem (without the ``d-") has been descri
 bed by some as ``marvelous".\n
LOCATION:https://researchseminars.org/talk/CANT2025/55/
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