3,642 research outputs found
Controlling iterated jumps of solutions to combinatorial problems
Among the Ramsey-type hierarchies, namely, Ramsey's theorem, the free set,
the thin set and the rainbow Ramsey theorem, only Ramsey's theorem is known to
collapse in reverse mathematics. A promising approach to show the strictness of
the hierarchies would be to prove that every computable instance at level n has
a low_n solution. In particular, this requires effective control of iterations
of the Turing jump. In this paper, we design some variants of Mathias forcing
to construct solutions to cohesiveness, the Erdos-Moser theorem and stable
Ramsey's theorem for pairs, while controlling their iterated jumps. For this,
we define forcing relations which, unlike Mathias forcing, have the same
definitional complexity as the formulas they force. This analysis enables us to
answer two questions of Wei Wang, namely, whether cohesiveness and the
Erdos-Moser theorem admit preservation of the arithmetic hierarchy, and can be
seen as a step towards the resolution of the strictness of the Ramsey-type
hierarchies.Comment: 32 page
A constructive commutative quantum Lovasz Local Lemma, and beyond
The recently proven Quantum Lovasz Local Lemma generalises the well-known
Lovasz Local Lemma. It states that, if a collection of subspace constraints are
"weakly dependent", there necessarily exists a state satisfying all
constraints. It implies e.g. that certain instances of the kQSAT quantum
satisfiability problem are necessarily satisfiable, or that many-body systems
with "not too many" interactions are always frustration-free.
However, the QLLL only asserts existence; it says nothing about how to find
the state. Inspired by Moser's breakthrough classical results, we present a
constructive version of the QLLL in the setting of commuting constraints,
proving that a simple quantum algorithm converges efficiently to the required
state. In fact, we provide two different proofs, one using a novel quantum
coupling argument, the other a more explicit combinatorial analysis. Both
proofs are independent of the QLLL. So these results also provide independent,
constructive proofs of the commutative QLLL itself, but strengthen it
significantly by giving an efficient algorithm for finding the state whose
existence is asserted by the QLLL. We give an application of the constructive
commutative QLLL to convergence of CP maps.
We also extend these results to the non-commutative setting. However, our
proof of the general constructive QLLL relies on a conjecture which we are only
able to prove in special cases.Comment: 43 pages, 2 conjectures, no figures; unresolved gap in the proof; see
arXiv:1311.6474 or arXiv:1310.7766 for correct proofs of the symmetric cas
Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory
The goal of this present manuscript is to introduce the reader to the
nonstandard method and to provide an overview of its most prominent
applications in Ramsey theory and combinatorial number theory.Comment: 126 pages. Comments welcom
Sampling-based proofs of almost-periodicity results and algorithmic applications
We give new combinatorial proofs of known almost-periodicity results for
sumsets of sets with small doubling in the spirit of Croot and Sisask, whose
almost-periodicity lemma has had far-reaching implications in additive
combinatorics. We provide an alternative (and L^p-norm free) point of view,
which allows for proofs to easily be converted to probabilistic algorithms that
decide membership in almost-periodic sumsets of dense subsets of F_2^n.
As an application, we give a new algorithmic version of the quasipolynomial
Bogolyubov-Ruzsa lemma recently proved by Sanders. Together with the results by
the last two authors, this implies an algorithmic version of the quadratic
Goldreich-Levin theorem in which the number of terms in the quadratic Fourier
decomposition of a given function is quasipolynomial in the error parameter,
compared with an exponential dependence previously proved by the authors. It
also improves the running time of the algorithm to have quasipolynomial
dependence instead of an exponential one.
We also give an application to the problem of finding large subspaces in
sumsets of dense sets. Green showed that the sumset of a dense subset of F_2^n
contains a large subspace. Using Fourier analytic methods, Sanders proved that
such a subspace must have dimension bounded below by a constant times the
density times n. We provide an alternative (and L^p norm-free) proof of a
comparable bound, which is analogous to a recent result of Croot, Laba and
Sisask in the integers.Comment: 28 page
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
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