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Nets Hawk Katz

Publications and source records attributed to Nets Hawk Katz.

At least 19 recordsLinked to original sources
arXiv2023

Kakeya sets from lines in $SL_2$

We prove that every Kakeya set in $\mathbb{R}^3$ formed from lines of the form $(a,b,0) + \operatorname{span}(c,d,1)$ with $ad-bc=1$ must have Hausdorff dimension $3$; Kakeya sets of this type are called $SL_2$ Kakeya sets. This result was also recently proved by Fässler and Orponen using different techniques. Our method combines induction on scales with a special structural property of $SL_2$ Kakeya sets, which says that locally such sets look like the pre-image of an arrangement of plane curves above a special type of map from $\mathbb{R}^3$ to $\mathbb{R}^2$, called a twisting projection. This reduces the study of $SL_2$ Kakeya sets to a Kakeya-type problem for plane curves; the latter is analyzed using a variant of Wolff's circular maximal function.

math.CA↗
arXiv2019

On the discretized sum-product problem

We give a new proof of the discretized ring theorem for sets of real numbers. As a special case, we show that if $A\subset\mathbb{R}$ is a $(δ,1/2)_1$-set in the sense of Katz and Tao, then either $A+A$ or $A.A$ must have measure at least $|A|^{1-\frac{1}{68}}$

math.CA↗
arXiv2019

A Kakeya maximal function estimate in four dimensions using planebrushes

We obtain an improved Kakeya maximal function estimate and improved Kakeya Hausdorff dimension estimate in $\mathbb{R}^4$ using a new geometric argument called the planebrush. A planebrush is a higher dimensional analogue of Wolff's hairbrush, which gives effective control on the size of Besicovitch sets when the lines through a typical point concentrate into a plane. When Besicovitch sets do not have this property, the existing trilinear estimates of Guth-Zahl can be used to bound the size of a Besicovitch set. In particular, we establish a maximal function estimate in $\mathbb{R}^4$ at dimension 3.049, and we prove that every Besicovitch set in $\mathbb{R}^4$ must have Hausdorff dimension at least 3.059.

math.CA↗
arXiv2018

On the polynomial Wolff axioms

We confirm a conjecture of Guth concerning the maximal number of $δ$-tubes, with $δ$-separated directions, contained in the $δ$-neighborhood of a real algebraic variety. Modulo a factor of $δ^{-\varepsilon}$, we also prove Guth and Zahl's generalized version for semialgebraic sets. Although the applications are to be found in harmonic analysis, the proof will employ deep results from algebraic and differential geometry, including Tarski's projection theorem and Gromov's algebraic lemma.

math.CA↗
arXiv2018

An improved bound on the Hausdorff dimension of Besicovitch sets in $\mathbb{R}^3$

We prove that any Besicovitch set in $\mathbb{R}^3$ must have Hausdorff dimension at least $5/2+ε_0$ for some small constant $ε_0>0$. This follows from a more general result about the volume of unions of tubes that satisfy the Wolff axioms. Our proof grapples with a new "almost counter example" to the Kakeya conjecture, which we call the $SL_2$ example; this object resembles a Besicovitch set that has Minkowski dimension 3 but Hausdorff dimension $5/2$. We believe this example may be an interesting object for future study.

math.CA↗
arXiv2014

Approximate aggregation in the neoclassical growth model with ideosyncratic shocks

We provide an explicit aggregation in the neoclassical growth model with aggregate shocks and uninsurable employment risk. We show there are two restrictions on the unemployment shock for approximate aggregation to occur. First the probability of unemployment must be positive for each agent in each time period. That ensures a strong precautionary savings motive. Second, we must have like agents having similar future prospects. That is agents with similar employment status and wealth must have similar employment paths. The solution of the model must have distribution of wealth as a state variable and hence the curse of dimensionality must be confronted. We sidestep this thorny issue by introducing a Walrassian auctioneer that communicates the optimal amount of invested in every period for every outcome of the shocks to the agents.

math.OC↗
arXiv2014

The flecnode polynomial: a central object in incidence geometry

We give a brief exposition of the proof of the Cayley-Salmon theorem and its recent role in incidence geometry. Even when we don't use the properties of ruled surfaces explicitly, the regime in which we have interesting results in point-line incidence problems often coincides with the regime in which lines are organized into ruled surfaces.

math.CO↗
arXiv2012

A note on the slightly supercritical Navier Stokes equations in the plane

We produce a new proof of Tao's result on the slightly supercritical Navier Stokes equations. Our proof has the advantage that it works in the plane while Tao's proof works only in dimensions three and higher. We accomplish this by studying the problem as a system of differential inequalities on the $L^2$ norms of the Littlewood Paley decomposition, along the lines of Pavlovic's proof of the Beale-Kato-Majda theorem.

math.AP↗
arXiv2011

On the Erdos distinct distance problem in the plane

In this paper, we prove that a set of $N$ points in ${\bf R}^2$ has at least $c{N \over \log N}$ distinct distances, thus obtaining the sharp exponent in a problem of Erdös. We follow the set-up of Elekes and Sharir which, in the spirit of the Erlangen program, allows us to study the problem in the group of rigid motions of the plane. This converts the problem to one of point-line incidences in space. We introduce two new ideas in our proof. In order to control points where many lines are incident, we create a cell decompostion using the polynomial ham sandwich theorem. This creates a dichotomy: either most of the points are in the interiors of the cells, in which case we immediately get sharp results, or alternatively the points lie on the walls of the cells, in which case they are in the zero set of a polynomial of suprisingly low degree, and we may apply the algebraic method. In order to control points where only two lines are incident, we use the flecnode polynomial of the Rev. George Salmon to conclude that most of the lines lie on a ruled surface. Then we use the geometry of ruled surfaces to complete the proof.

math.CO↗
arXiv2011

Structure in additively nonsmoothing sets

Sets with many additive quadruples are guaranteed to have many additive octuples, by Hölder's inequality. Sets with not many more than this are said to be additively nonsmoothing. We give a new proof of a structural theorem for nonsmoothing sets that originally appeared in work of the authors (\cite{BK}) on the size of cap sets in $F_3 ^N$.

math.CO↗
arXiv2011

New Bounds on cap sets

We provide an improvement over Meshulam's bound on cap sets in $F_3^N$. We show that there exist universal $ε>0$ and $C>0$ so that any cap set in $F_3^N$ has size at most $C {3^N \over N^{1+ε}}$. We do this by obtaining quite strong information about the additive combinatorial properties of the large spectrum.

math.CA↗
arXiv2009

On Freiman's Theorem in Nilpotent Groups

We generalize a result of Tao which extends Freiman's theorem to the Heisenberg group. We extend it to simply connected nilpotent Lie groups of arbitrary step.

math.CO↗
arXiv2008

Algebraic Methods in Discrete Analogs of the Kakeya Problem

We prove the joints conjecture, showing that for any $N$ lines in ${\Bbb R}^3$, there are at most $O(N^{3 \over 2})$ points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given $N^2$ lines in ${\Bbb R}^3$ so that no $N$ lines lie in the same plane and so that each line intersects a set $P$ of points in at least $N$ points then the cardinality of the set of points is $Ω(N^3)$. Both our proofs are adaptations of Dvir's argument for the finite field Kakeya problem.

math.CO↗
arXiv2008

On additive doubling and energy

We show that if A is a set having small subtractive doubling in an abelian group, that is |A-A|< K|A|, then there is a polynomially large subset B of A-A so that the additive energy of B is large than (1/K)^{1 - ε) where epsilon is a positive, universal exponent. (1/37 seems to suffice.)

math.CO↗
arXiv2007

Garaev's Inequality in finite fields not of prime order

We prove a version of Garaev's sum product theorem in the set of finite fields with non-prime order. Because of the presence of subfields, this seems to require some hypotheses on the set. We work under a condition analogous to having Hausdorff dimension less than 1/2. Under these conditions, we obtain a sum-product theorem with exponent 49/48.

math.NT↗