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Ambrus Gergely: On the density of planar sets avoiding unit distances

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Csütörtök, 13. Október 2022, 12:30 - 13:30
Abstract: We prove a 50-year-old conjecture of Moser and Erdős stating
that the density of any measurable planar set not containing two points
at unit distance is less than 1/4. We utilize Fourier analytic and
linear programming methods in order to prove the upper bound of 0.247,
which improves on the previous strongest estimate 0.254 reached by
Ambrus and Matolcsi (2022), and gets considerably closer to the
conjectured optimal lower bound of 0.229 which follows from a
construction of Croft (1969). The improvement is due to two factors: on
the one hand, refining and polishing the theoretical background, while
on the other hand, utilizing high complexity computer search implemented
on large-scale computers which results in a set of linear constraints
stemming from a 23-vertex graph. Our approach also entails the method of
estimating fractional chromatic numbers, for which a further
generalization and a related conjecture is given.
This is a joint work with A. Csiszárik, M. Matolcsi, D. Varga and P.
Zsámboki.

További információk:
http://www.math.u-szeged.hu/Geo/_site/index.php/seminar-blog
 
Az előadást Zoom-on is közvetítjük: https://us06web.zoom.us/j/82507865705 
 

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