273 research outputs found
Powers of sequences and recurrence
We study recurrence, and multiple recurrence, properties along the -th
powers of a given set of integers. We show that the property of recurrence for
some given values of does not give any constraint on the recurrence for the
other powers. This is motivated by similar results in number theory concerning
additive basis of natural numbers. Moreover, motivated by a result of Kamae and
Mend\`es-France, that links single recurrence with uniform distribution
properties of sequences, we look for an analogous result dealing with higher
order recurrence and make a related conjecture.Comment: 30 pages. Numerous small changes made. To appear in the Proceedings
of the London Mathematical Societ
Under recurrence in the Khintchine recurrence theorem
The Khintchine recurrence theorem asserts that on a measure preserving
system, for every set and , we have for infinitely many . We show that there
are systems having under-recurrent sets , in the sense that the inequality
holds for every . In
particular, all ergodic systems of positive entropy have under-recurrent sets.
On the other hand, answering a question of V.~Bergelson, we show that not all
mixing systems have under-recurrent sets. We also study variants of these
problems where the previous strict inequality is reversed, and deduce that
under-recurrence is a much more rare phenomenon than over-recurrence. Finally,
we study related problems pertaining to multiple recurrence and derive some
interesting combinatorial consequences.Comment: 18 pages. Referee's comments incorporated. To appear in the Israel
Journal of Mathematic
Töredékek a térben tér a töredékekben
Tizennyolc Ă©ve foglalkozom az esztergomi kirĂĄlyi palota festett
töredékeivel, a kirålyi kåpolnåval és hozzå tartozó helyiségeivel, valamint a
Studioloval. A kĂĄpolna kĂŒlönösen a szĂvemhez nĆtt, gyönyörƱ darabjaival,
összetettségével és az ezzel jåró végelåthatatlan feladatokkal. Nem hinném,
hogy lĂ©tezik ma Ă©lĆ ember, aki jobban ismernĂ© minden zegĂ©t-zĂșgĂĄt, az összes
kicsi és nagyobb bajåval, négyzetcentiméterek megható fordulataival, a
âtalpĂĄtĂłlâ egĂ©szen a âfeje bĂșbjĂĄigâ. Ez a tĂ©r, elfogadja hiĂĄnyossĂĄgait, nem törekszik a âmindenĂĄron egĂ©szreâ,
vagy kiegĂ©szĂtettre.Ebben az egysĂ©gben a kiegĂ©szĂtett rekonstrukciĂłs felĂŒletek is Ășj funkciĂłt
kapnak, a fragmentĂĄlt jelenetekkel valami egĂ©sszĂ© formĂĄlĂłdva, egy Ășj
mƱalkotåsså ållnak össze: a Magyar kirålyi kåpolna- töredékké
Random differences in Szemerédi's theorem and related results
à paraßtre dans le Journal d'Analyse MathématiqueWe introduce a new, elementary method for studying random differences in arithmetic progressions and convergence phenomena along random sequences of integers. We apply our method to obtain significant improvements on two results. The first one concerns existence of arithmetic progresseions of given length in any set of integers of positive density, with differences in a random subset of the integers. The second one concerns almost everywhere convergence of double ergodic averages along partially random sequences
Random Sequences and Pointwise Convergence of Multiple Ergodic Averages
International audienceWe prove pointwise convergence, as , for the multiple ergodic averages , where and are commuting measure preserving transformations, and is a random version of the sequence for some appropriate . We also prove similar mean convergence results for averages of the form , as well as pointwise results when and are powers of the same transformation. The deterministic versions of these results, where one replaces with , remain open, and we hope that our method will indicate a fruitful way to approach these problems as well
Additive bases arising from functions in a Hardy field
A classical additive basis question is Waring's problem. It has been extended
to integer polynomial and non-integer power sequences. In this paper, we will
consider a wider class of functions, namely functions from a Hardy field, and
show that they are asymptotic bases.Comment: 11 pages, reference problem fixe
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