BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Jorg Brudern (Universitat Gottingen)
DTSTART:20250521T163000Z
DTEND:20250521T165500Z
DTSTAMP:20260423T024834Z
UID:CANT2025/65
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CANT2025/65/
 ">Expander estimates for cubes</a>\nby Jorg Brudern (Universitat Gottingen
 ) as part of Combinatorial and additive number theory (CANT 2025)\n\nLectu
 re held in CUNY Graduate Center - Science Center (4th floor).\n\nAbstract\
 nSuppose that $\\mathcal A$ is a subset of the natural numbers. The suprem
 um $\\alpha$ of all $t$ with \n$$ \\limsup N^{-t} \\#\\{a\\in{\\mathcal A}
 : a\\le N\\} >0 $$\nis the {\\em exponential density} of $\\mathcal A$.\n\
 nWe examine what happens if one adds a power to $\\mathcal A$. Fix $k\\ge 
 2$\, and let $\\beta_k$ be the exponential density of\n$$ \\{ x^k+a : x\\i
 n {\\mathbb N}\, \\\, a\\in{\\mathcal A}\\}.$$\nIt is easy to see that $\\
 beta_2= \\min (1\, \\frac12 +\\alpha).$ One might guess that\n$$ \\beta_k 
 = \\min (1\, \\frac{1}{k}+\\alpha) \\eqno (*) $$\nholds for all $k$\, but 
 we are far from a proof. All current world records for this problem are du
 e to Davenport\, and are 80 years old. In this interim report on ongoing w
 ork with Simon Myerson\, we describe a method \nfor $k=3$ that improves Da
 venport's results when $\\alpha>3/5$\, and that confirms (*) in an interva
 l $(\\alpha_0\, 1]$. A concrete value for $\\alpha_0$ will be released dur
 ing the talk\, and if time permits\, we also discuss the perspectives to g
 eneralize the approach to larger values of $k$.\n
LOCATION:https://researchseminars.org/talk/CANT2025/65/
END:VEVENT
END:VCALENDAR