944 research outputs found
Exceptional sets for Diophantine inequalities
We apply Freeman's variant of the Davenport-Heilbronn method to investigate
the exceptional set of real numbers not close to some value of a given real
diagonal form at an integral argument. Under appropriate conditions, we show
that the exceptional set in the interval [-N,N] has measure O(N^{1-c}), for a
positive number c
The Average-Case Area of Heilbronn-Type Triangles
From among triangles with vertices chosen from points in
the unit square, let be the one with the smallest area, and let be the
area of . Heilbronn's triangle problem asks for the maximum value assumed by
over all choices of points. We consider the average-case: If the
points are chosen independently and at random (with a uniform distribution),
then there exist positive constants and such that for all large enough values of , where is the expectation of
. Moreover, , with probability close to one. Our proof
uses the incompressibility method based on Kolmogorov complexity; it actually
determines the area of the smallest triangle for an arrangement in ``general
position.''Comment: 13 pages, LaTeX, 1 figure,Popular treatment in D. Mackenzie, On a
roll, {\em New Scientist}, November 6, 1999, 44--4
A Diophantine problem with prime variables
We study the distribution of the values of the form , where , and
are non-zero real number not all of the same sign, with irrational, and , and are prime numbers. We prove
that, when , these value approximate rather closely any
prescribed real number.Comment: submitte
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