Mark Lewko
Orthonormal Systems in Linear Spans.
Mon, 05/14/2012 - 15:01 — SomNambulAuthors: Allison Lewko, Mark Lewko
Subjects: Classical Analysis and ODEs
AbstractWe show that any $N$-dimensional linear subspace of $L^2(\mathbb{T})$ admits
an orthonormal system such that the $L^2$ norm of the square variation operator
$V^2$ is as small as possible. When applied to the span of the trigonometric
system, we obtain an orthonormal system of trigonometric polynomials with a
$V^2$ operator that is considerably smaller than the associated operator for
the trigonometric system itself.An Exact Asymptotic for the Square Variation of Partial Sum Processes.
Tue, 06/07/2011 - 15:01 — SomNambulAuthors: Allison Lewko, Mark Lewko
Subjects: Probability
AbstractWe establish an exact asymptotic formula for the square variation of certain
partial sum processes. Let $\{X_{i}\}$ be a sequence of independent,
identically distributed mean zero random variables with finite variance
$\sigma$ and satisfying a moment condition $\mathbb{E}[|X_{i}|^{2+\delta} ] <
\infty$ for some $\delta > 0$. If we let $\mathcal{P}_{N}$ denote the set of
all possible partitions of the interval $[N]$ into subintervals, then we have
that $\max_{\pi \in \mathcal{P}_{N}} \sum_{I \in \pi} | \sum_{i\in I} X_{i}|^2
\sim 2 \sigma^2N \ln \ln(N)$ holds almost surely.An Improved Upper Bound for the Sum-free Subset Constant.
Mon, 07/19/2010 - 15:01 — SomNambulAuthors: Mark Lewko
Subjects: Combinatorics
AbstractWe show that the optimal constant in Erd\"{o}s' sum-free subset theorem
cannot be larger than $11/28 \approx .393$.Sets of Large Doubling and a Question of Rudin.
Thu, 03/25/2010 - 16:01 — SomNambulAuthors: Allison Lewko, Mark Lewko
Subjects: Classical Analysis and ODEs
AbstractWe construct a $\Lambda(4)$ set which is not a finite union of $B_2[G]$ sets,
answering a question of Rudin. Our construction is an interesting combinatorial
object in its own right, and provides a counterexample to a weaker
characterization of $\Lambda(4)$ sets than stated in Rudin's original question.
It also serves as a counterexample to several 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.
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