9,703 research outputs found
Cokernels of random matrices satisfy the Cohen-Lenstra heuristics
Let A be an n by n random matrix with iid entries taken from the p-adic
integers or Z/NZ. Then under mild non-degeneracy conditions the cokernel of A
has a universal probability distribution. In particular, the p-part of an iid
random matrix over the integers has cokernel distributed according to the
Cohen-Lenstra measure up to an exponentially small error.Comment: 21 pages; submitte
Products of Differences over Arbitrary Finite Fields
There exists an absolute constant such that for all and all
subsets of the finite field with elements, if
, then Any suffices for sufficiently large
. This improves the condition , due to Bennett, Hart,
Iosevich, Pakianathan, and Rudnev, that is typical for such questions.
Our proof is based on a qualitatively optimal characterisation of sets for which the number of solutions to the equation is nearly
maximum.
A key ingredient is determining exact algebraic structure of sets for
which is nearly minimum, which refines a result of Bourgain and
Glibichuk using work of Gill, Helfgott, and Tao.
We also prove a stronger statement for when are sets in a prime field,
generalising a result of Roche-Newton, Rudnev, Shkredov, and the authors.Comment: 42 page
Note on the Theory of Correlation Functions
The purpose of this note is to improve the current theoretical results for
the correlation functions of the Mobius sequence and the
Liouville sequence .Comment: Sixty Six Pages. Keywords: Autocorrelation function, Correlation
function, Multiplicative function, Liouville function, Mobius function, von
Mangoldt function, Exponential Su
General Strong Polarization
Arikan's exciting discovery of polar codes has provided an altogether new way
to efficiently achieve Shannon capacity. Given a (constant-sized) invertible
matrix , a family of polar codes can be associated with this matrix and its
ability to approach capacity follows from the {\em polarization} of an
associated -bounded martingale, namely its convergence in the limit to
either or . Arikan showed polarization of the martingale associated with
the matrix to get
capacity achieving codes. His analysis was later extended to all matrices
that satisfy an obvious necessary condition for polarization.
While Arikan's theorem does not guarantee that the codes achieve capacity at
small blocklengths, it turns out that a "strong" analysis of the polarization
of the underlying martingale would lead to such constructions. Indeed for the
martingale associated with such a strong polarization was shown in two
independent works ([Guruswami and Xia, IEEE IT '15] and [Hassani et al., IEEE
IT '14]), resolving a major theoretical challenge of the efficient attainment
of Shannon capacity.
In this work we extend the result above to cover martingales associated with
all matrices that satisfy the necessary condition for (weak) polarization. In
addition to being vastly more general, our proofs of strong polarization are
also simpler and modular. Specifically, our result shows strong polarization
over all prime fields and leads to efficient capacity-achieving codes for
arbitrary symmetric memoryless channels. We show how to use our analyses to
achieve exponentially small error probabilities at lengths inverse polynomial
in the gap to capacity. Indeed we show that we can essentially match any error
probability with lengths that are only inverse polynomial in the gap to
capacity.Comment: 73 pages, 2 figures. The final version appeared in JACM. This paper
combines results presented in preliminary form at STOC 2018 and RANDOM 201
Moments and distribution of central L-values of quadratic twists of elliptic curves
We show that if one can compute a little more than a particular moment for
some family of L-functions, then one has upper bounds of the conjectured order
of magnitude for all smaller (positive, real) moments and a one-sided central
limit theorem holds. We illustrate our method for the family of quadratic
twists of an elliptic curve, obtaining sharp upper bounds for all moments below
the first. We also establish a one sided central limit theorem supporting a
conjecture of Keating and Snaith. Our work leads to a conjecture on the
distribution of the order of the Tate-Shafarevich group for rank zero quadratic
twists of an elliptic curve, and establishes the upper bound part of this
conjecture (assuming the Birch-Swinnerton-Dyer conjecture).Comment: 28 page
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