24,109 research outputs found
On a conjecture of Brown concerning accessible sets
AbstractIn this note we use a sequence constructed by Furstenberg in 1981 to disprove the following conjecture posed by Brown: If a set of positive numbers L is such that for any finite coloring of N there are arbitrarily long monochromatic sequences of distinct integers with all gaps in L, then for any finite coloring of N there are arbitrarily long monochromatic arithmetic progressions whose common differences belong to L
Translation invariance, exponential sums, and Waring's problem
We describe mean value estimates for exponential sums of degree exceeding 2
that approach those conjectured to be best possible. The vehicle for this
recent progress is the efficient congruencing method, which iteratively
exploits the translation invariance of associated systems of Diophantine
equations to derive powerful congruence constraints on the underlying
variables. There are applications to Weyl sums, the distribution of polynomials
modulo 1, and other Diophantine problems such as Waring's problem.Comment: Submitted to Proceedings of the ICM 201
Chaotic Orbits in Thermal-Equilibrium Beams: Existence and Dynamical Implications
Phase mixing of chaotic orbits exponentially distributes these orbits through
their accessible phase space. This phenomenon, commonly called ``chaotic
mixing'', stands in marked contrast to phase mixing of regular orbits which
proceeds as a power law in time. It is operationally irreversible; hence, its
associated e-folding time scale sets a condition on any process envisioned for
emittance compensation. A key question is whether beams can support chaotic
orbits, and if so, under what conditions? We numerically investigate the
parameter space of three-dimensional thermal-equilibrium beams with space
charge, confined by linear external focusing forces, to determine whether the
associated potentials support chaotic orbits. We find that a large subset of
the parameter space does support chaos and, in turn, chaotic mixing. Details
and implications are enumerated.Comment: 39 pages, including 14 figure
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
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