1,701 research outputs found
Large subsets of discrete hypersurfaces in contain arbitrarily many collinear points
In 1977 L.T. Ramsey showed that any sequence in with bounded
gaps contains arbitrarily many collinear points. Thereafter, in 1980, C.
Pomerance provided a density version of this result, relaxing the condition on
the sequence from having bounded gaps to having gaps bounded on average. We
give a higher dimensional generalization of these results. Our main theorem is
the following.
Theorem: Let , let be a
Lipschitz map and let have positive upper Banach
density. Then contains arbitrarily many collinear points.
Note that Pomerance's theorem corresponds to the special case . In our
proof, we transfer the problem from a discrete to a continuous setting,
allowing us to take advantage of analytic and measure theoretic tools such as
Rademacher's theorem.Comment: 16 pages, small part of the argument clarified in light of
suggestions from the refere
Multiplicative combinatorial properties of return time sets in minimal dynamical systems
We investigate the relationship between the dynamical properties of minimal
topological dynamical systems and the multiplicative combinatorial properties
of return time sets arising from those systems. In particular, we prove that
for a residual sets of points in any minimal system, the set of return times to
any non-empty, open set contains arbitrarily long geometric progressions. Under
the separate assumptions of total minimality and distality, we prove that
return time sets have positive multiplicative upper Banach density along
and along multiplicative subsemigroups of ,
respectively. The primary motivation for this work is the long-standing open
question of whether or not syndetic subsets of the positive integers contain
arbitrarily long geometric progressions; our main result is some evidence for
an affirmative answer to this question.Comment: 32 page
Single and multiple recurrence along non-polynomial sequences
We establish new recurrence and multiple recurrence results for a rather
large family of non-polynomial functions which includes tempered
functions defined in [11], as well as functions from a Hardy field with the
property that for some , and . Among
other things, we show that for any , any invertible
probability measure preserving system , any
with , and any , the sets of returns and possess somewhat unexpected properties of
largeness; in particular, they are thick, i.e., contain arbitrarily long
intervals.Comment: 51 page
Disjointness for measurably distal group actions and applications
We generalize Berg's notion of quasi-disjointness to actions of countable
groups and prove that every measurably distal system is quasi-disjoint from
every measure preserving system. As a corollary we obtain easy to check
necessary and sufficient conditions for two systems to be disjoint, provided
one of them is measurably distal. We also obtain a Wiener--Wintner type theorem
for countable amenable groups with distal weights and applications to weighted
multiple ergodic averages and multiple recurrence.Comment: 28 page
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