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Richard Oberlin

Publications and source records attributed to Richard Oberlin.

At least 19 recordsLinked to original sources
arXiv2015

Dimensions of sums with self-similar sets

For some self-similar sets K in d-dimensional Euclidean space we obtain certain lower bounds for the lower Minkowski dimension of K+E in terms of the lower Minkowski dimension of E.

math.CA↗
arXiv2015

Variation-norm and fluctuation estimates for ergodic bilinear averages

For any dynamical system, we show that higher variation-norms for the sequence of ergodic bilinear averages of two functions satisfy a large range of bilinear Lp estimates. It follows that, with probability one, the number of fluctuations along this sequence may grow at most polynomially with respect to (the growth of) the underlying scale. These results strengthen previous works of Lacey and Bourgain where almost surely convergence of the sequence was proved (which is equivalent to the qualitative statement that the number of fluctuations is finite at each scale). Via transference, the proof reduces to establishing new bilinear Lp bounds for variation-norms of truncated bilinear operators on R, and the main ingredient of the proof of these bounds is a variation-norm extension of maximal Bessel inequalities of Lacey and Demeter--Tao--Thiele.

math.CA↗
arXiv2014

Unions of lines in F^n

We show that if a collection of lines in a vector space over a finite field has "dimension" at least 2(d-1) + beta, then its union has "dimension" at least d + beta. This is the sharp estimate of its type when no structural assumptions are placed on the collection of lines. We also consider some refinements and extensions of the main result, including estimates for unions of k-planes.

math.CA↗
arXiv2012

Unit Distance Problems

We study some discrete and continuous variants of the following problem of Erdos: given a finite subset P of R^2 or R^3, what is the maximum number of pairs (p_1,p_2) with p_1,p_2 in P and |p_1 -p_2 |=1?

math.CA↗
arXiv2012

Variational bounds for a dyadic model of the bilinear Hilbert transform

We prove variation-norm estimates for the Walsh model of the truncated bilinear Hilbert transform, extending related results of Lacey, Thiele, and Demeter. The proof uses analysis on the Walsh phase plane and two new ingredients: (i) a variational extension of a lemma of Bourgain by Nazarov-Oberlin-Thiele, and (ii) a variation-norm Rademacher-Menshov theorem of Lewko-Lewko.

math.CA↗
arXiv2011

Estimates for compositions of maximal operators with singular integrals

We prove weak-type (1,1) estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator $Δ^*Ψ$ where $Δ^*$ is Bourgain's maximal multiplier operator and $Ψ$ is the sum of several modulated singular integrals; here our method yields a significantly improved bound for the $L^q$ operator norm when $1 < q < 2$. We also consider associated variation-norm estimates.

math.CA↗
arXiv2010

A variation norm Carleson theorem

We strengthen the Carleson-Hunt theorem by proving $L^p$ estimates for the $r$-variation of the partial sum operators for Fourier series and integrals, for $p>\max\{r',2\}$. Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

math.CA↗
arXiv2010

New uniform bounds for a Walsh model of the bilinear Hilbert transform

We prove old and new $L^p$ bounds for the quartile operator, a Walsh model of the bilinear Hilbert transform, uniformly in the parameter that models degeneration of the bilinear Hilbert transform. We obtain the full range of exponents that can be expected from known bounds in the degenerate and non-degenerate cases. For the new estimates with exponents p close to 1 the argument relies on a multi-frequency Calderon-Zygmund decomposition.

math.CA↗
arXiv2009

The Kakeya set and maximal conjectures for algebraic varieties over finite fields

Using the polynomial method of Dvir \cite{dvir}, we establish optimal estimates for Kakeya sets and Kakeya maximal functions associated to algebraic varieties $W$ over finite fields $F$. For instance, given an $n-1$-dimensional projective variety $W \subset ¶^n(F)$, we establish the Kakeya maximal estimate $$ \| \sup_{γ\ni w} \sum_{v \in γ(F)} |f(v)| \|_{\ell^n(W)} \leq C_{n,W,d} |F|^{(n-1)/n} \|f\|_{\ell^n(F^n)}$$ for all functions $f: F^n \to \R$ and $d \geq 1$, where for each $w \in W$, the supremum is over all irreducible algebraic curves in $F^n$ of degree at most $d$ that pass through $w$ but do not lie in $W$, and with $C_{n,W,d}$ depending only on $n, d$ and the degree of $W$; the special case when $W$ is the hyperplane at infinity in particular establishes the Kakeya maximal function conjecture in finite fields, which in turn strengthens the results of Dvir.

math.CO↗
arXiv2006

Two bounds for the x-ray transform

We use the arithmetic-combinatorial method of Katz and Tao to give mixed-norm estimates for the x-ray transform on R^d when d \geq 4. As an application, we obtain an improved estimate for the Hausdorff dimension of (d,k) sets, which are subsets of R^d containing a translate of every k-plane.

math.CA↗
arXiv2005

Bounds for Kakeya-type maximal operators associated with k-planes

A (d,k) set is a subset of R^d containing a translate of every k-dimensional plane. Bourgain showed that for k \geq k_{cr}(d), where k_{cr}(d) solves 2^{k_{cr}-1}+k_{cr} = d, every (d,k) set has positive Lebesgue measure. We give a short proof of this result which allows for an improved L^p estimate of the corresponding maximal operator, and which demonstrates that a lower value of k_{cr} could be obtained if improved mixed-norm estimates for the x-ray transform were known.

math.CA↗