88,351 research outputs found
The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures
We study the asymptotic behaviour of higher order correlations as a function of the
parameters and , where are bounded multiplicative
functions, are integer shifts, and is large. Our main
structural result asserts, roughly speaking, that such correlations
asymptotically vanish for almost all if does not (weakly)
pretend to be a twisted Dirichlet character , and
behave asymptotically like a multiple of otherwise. This
extends our earlier work on the structure of logarithmically averaged
correlations, in which the parameter is averaged out and one can set .
Among other things, the result enables us to establish special cases of the
Chowla and Elliott conjectures for (unweighted) averages at almost all scales;
for instance, we establish the -point Chowla conjecture for odd or equal to for
all scales outside of a set of zero logarithmic density.Comment: 48 pages, no figures. Referee comments incorporate
On barycentric subdivision, with simulations
Consider the barycentric subdivision which cuts a given triangle along its
medians to produce six new triangles. Uniformly choosing one of them and
iterating this procedure gives rise to a Markov chain. We show that almost
surely, the triangles forming this chain become flatter and flatter in the
sense that their isoperimetric values goes to infinity with time. Nevertheless,
if the triangles are renormalized through a similitude to have their longest
edge equal to [0,1]\subset\CC (with 0 also adjacent to the shortest edge),
their aspect does not converge and we identify the limit set of the opposite
vertex with the segment [0,1/2]. In addition we prove that the largest angle
converges to in probability. Our approach is probabilistic and these
results are deduced from the investigation of a limit iterated random function
Markov chain living on the segment [0,1/2]. The stationary distribution of this
limit chain is particularly important in our study. In an appendix we present
related numerical simulations (not included in the version submitted for
publication)
Apollonian structure in the Abelian sandpile
The Abelian sandpile process evolves configurations of chips on the integer
lattice by toppling any vertex with at least 4 chips, distributing one of its
chips to each of its 4 neighbors. When begun from a large stack of chips, the
terminal state of the sandpile has a curious fractal structure which has
remained unexplained. Using a characterization of the quadratic growths
attainable by integer-superharmonic functions, we prove that the sandpile PDE
recently shown to characterize the scaling limit of the sandpile admits certain
fractal solutions, giving a precise mathematical perspective on the fractal
nature of the sandpile.Comment: 27 Pages, 7 Figure
Alexandrov geometry: preliminary version no. 1
This is a preliminary version of our book. It goes up to the definition of
dimension, which is about 30% of the material we plan to include.
If you use it as a reference, do not forget to include the version number
since the numbering will be changed.Comment: 238 pages, 35 figure
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