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SUMMARY:Binary versus Ternary
DTSTART:20241118T100000Z
DTEND:20241118T110000Z
DTSTAMP:20241105T183700Z
UID:indico-event-12591@indico.math.cnrs.fr
CONTACT:regis.de-la-breteche@imj-prg.fr
DESCRIPTION:Speakers: Michael Drmota (TU Wien\, Autriche)\n\nThe relation
between the binary and the ternary expansion of a given positive integer o
r a class of integers is still not completely understood. For example\, we
know almost nothing about the binary expansion of powers of 3. Only recen
tly Spiegelhofer proved a "folklore conjecture" saying that there are infi
nitely many $n$ with $s_2(n) = s_3(n)$\, where $s_2(n)$ and $s_3(n)$ denot
e the binary and ternary sum-of-digits functions\, respectively.The purpos
e of this talk is to present a far reaching generalization of this result.
It is show that the set of pairs $(s_2(n)\,s_3(n))$ covers almost the who
le first quadrant of lattice points (only with possible gaps in the bounda
ry region). Interestingly the proof requires a combination of techniques f
rom analytic number theory (Gowers norms\, level-of-distribution results\,
exponential sums) and Diophantine approximation (Baker's theorem\, $p$-ad
ic subspace theorem).This is joint work with Lukas Spiegelhofer. \n\nhttp
s://indico.math.cnrs.fr/event/12591/
LOCATION:Salle Grisvard\, IHP\, Paris
URL:https://indico.math.cnrs.fr/event/12591/
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