35 research outputs found
On the Structure of Sets of Large Doubling
We investigate the structure of finite sets where is
large. We present a combinatorial construction that serves as a counterexample
to natural conjectures in the pursuit of an "anti-Freiman" theory in additive
combinatorics. In particular, we answer a question along these lines posed by
O'Bryant. Our construction also answers several questions about the nature of
finite unions of and sets, and enables us to construct
a set which does not contain large or
sets.Comment: 23 pages, changed title, revised version reflects work of Meyer that
we were previously unaware o
The Contest Between Simplicity and Efficiency in Asynchronous Byzantine Agreement
In the wake of the decisive impossibility result of Fischer, Lynch, and
Paterson for deterministic consensus protocols in the aynchronous model with
just one failure, Ben-Or and Bracha demonstrated that the problem could be
solved with randomness, even for Byzantine failures. Both protocols are natural
and intuitive to verify, and Bracha's achieves optimal resilience. However, the
expected running time of these protocols is exponential in general. Recently,
Kapron, Kempe, King, Saia, and Sanwalani presented the first efficient
Byzantine agreement algorithm in the asynchronous, full information model,
running in polylogarithmic time. Their algorithm is Monte Carlo and drastically
departs from the simple structure of Ben-Or and Bracha's Las Vegas algorithms.
In this paper, we begin an investigation of the question: to what extent is
this departure necessary? Might there be a much simpler and intuitive Las Vegas
protocol that runs in expected polynomial time? We will show that the
exponential running time of Ben-Or and Bracha's algorithms is no mere accident
of their specific details, but rather an unavoidable consequence of their
general symmetry and round structure. We define a natural class of "fully
symmetric round protocols" for solving Byzantine agreement in an asynchronous
setting and show that any such protocol can be forced to run in expected
exponential time by an adversary in the full information model. We assume the
adversary controls Byzantine processors for , where is an
arbitrary positive constant . We view our result as a step toward
identifying the level of complexity required for a polynomial-time algorithm in
this setting, and also as a guide in the search for new efficient algorithms.Comment: 21 page
Estimates for the Square Variation of Partial Sums of Fourier Series and their Rearrangements
We investigate the square variation operator (which majorizes the
partial sum maximal operator) on general orthonormal systems (ONS) of size .
We prove that the norm of the operator is bounded by on
any ONS. This result is sharp and refines the classical Rademacher-Menshov
theorem. We show that this can be improved to for the
trigonometric system, which is also sharp. We show that for any choice of
coefficients, this truncation of the trigonometric system can be rearranged so
that the norm of the associated operator is .
We also show that for , a bounded ONS of size can be rearranged so
that the norm of the operator is at most
uniformly for all choices of coefficients. This refines Bourgain's work on
Garsia's conjecture, which is equivalent to the case. Several
other results on operators of this form are also obtained. The proofs rely on
combinatorial and probabilistic methods.Comment: 37 pages, several minor edit
A Variational Barban-Davenport-Halberstam Theorem
We prove variational forms of the Barban-Davenport-Halberstam Theorem and the
large sieve inequality. We apply our result to prove an estimate for the sum of
the squares of prime differences, averaged over arithmetic progressions.Comment: 23 pages, some typos fixed, additional remarks added at end of pape