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SUMMARY:Density results and diophantine inequalities using the circle meth
 od - Paweł Nosal\, University of Warwick
DTSTART:20260306T160000Z
DTEND:20260306T170000Z
UID:TALK242941@talks.cam.ac.uk
CONTACT:Kelly Wang
DESCRIPTION:In 1929 Oppenheim asked whether the image $Q(\\mathbb{Z})$ of 
 an indefinite\, non-degenerate quadratic form $Q$ in more than $5$ variabl
 es\, which is not proportional to a rational quadratic form is dense in $\
 \mathbb{R}$. This question came to be known as the Oppenheim conjecture an
 d was answered in 1986 by Margulis\, who used methods coming from homogene
 ous dynamics to show that in fact $3$ variables suffice. Unfortunately\, t
 hese methods don't readily apply to hypersurfaces of higher degrees. \n\nI
 n 1946\, Davenport and Heilbronn used the duality between $\\mathbb{R}$ an
 d itself and introduced a new variant of the circle method\, which allowed
  them to prove that the integer image of a special polynomial is dense in 
 $\\mathbb{R}$. We present a refinement of their method by Freeman and show
  how it can be applied to solve counting problems connected to denseness r
 esults and Diophantine inequalities. This talk is intended for an audience
  with no prior experience with circle method and will include a gentle int
 roduction to classical theory.
LOCATION:MR13
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