1,235 research outputs found
On distinct cross-ratios and related growth problems
It is shown that for a finite set of four or more complex numbers, the
cardinality of the set of all cross-ratios generated by quadruples of
pair-wise distinct elements of is and without the logarithmic
factor in the real case. The set always grows under both addition and
multiplication. The cross-ratio arises, in particular, in the study of the open
question of the minimum number of triangle areas, with two vertices in a given
non-collinear finite point set in the plane and the third one at the fixed
origin. The above distinct cross-ratio bound implies a new lower bound for the
latter question, and enables one to show growth of the set
under multiplication. It seems
reasonable to conjecture that more-fold product, as well as sum sets of this
set or continue growing ad infinitum.Comment: 9p
On distance measures for well-distributed sets
In this paper we investigate the Erd\"os/Falconer distance conjecture for a
natural class of sets statistically, though not necessarily arithmetically,
similar to a lattice. We prove a good upper bound for spherical means that have
been classically used to study this problem. We conjecture that a majorant for
the spherical means suffices to prove the distance conjecture(s) in this
setting. For a class of non-Euclidean distances, we show that this generally
cannot be achieved, at least in dimension two, by considering integer point
distributions on convex curves and surfaces. In higher dimensions, we link this
problem to the question about the existence of smooth well-curved hypersurfaces
that support many integer points
Integrability versus topology of configuration manifolds and domains of possible motions
We establish a generic sufficient condition for a compact -dimensional
manifold bearing an integrable geodesic flow to be the -torus. As a
complementary result, we show that in the case of domains of possible motions
with boundary, the first Betti number of the domain of possible motions may be
arbitrarily large
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