arXiv subjects

Arian Nadjimzadah

Publications and source records attributed to Arian Nadjimzadah.

9 recordsLinked to original sources

arXiv2026

Lifting curved Kakeya sets to linear Kakeya sets

We prove that many curved Kakeya problems, originally motivated by Hörmander's oscillatory integral problem, lift to the classical Kakeya problem in higher dimensions. As a consequence, if the classical Kakeya set conjecture were true in all dimensions, the families of curves whose curved Kakeya sets have full dimension are dense among Hörmander-type families. This is in contrast to the nowhere dense Bourgain's condition, which is a necessary condition for best-case curved Kakeya maximal function estimates. Furthermore the linear Kakeya set conjecture would imply a fairly complete understanding of Kakeya sets of quadratic curves, as first studied systematically by Wisewell. These results give a better understanding of a question posed by Guo--Guth--Nadjimzadah--Shen--Zhang, and they more generally show that the classical Kakeya conjecture in high dimensions rests on a broad range of curved Kakeya problems in lower dimensions.

math.CA↗

arXiv2026

New curved Kakeya estimates

We prove that for an open class of translation-invariant Hörmander-type phase functions, the corresponding curved Kakeya sets in $\mathbb R^3$ have Hausdorff dimension at least $(13-\sqrt{13})/4 = 2.348\ldots$. Previous methods, such as polynomial partitioning, have attained at best dimension $2+1/3$ for curved Kakeya sets in $\mathbb R^3$. The proof combines Wolff's classical hairbrush argument with a new Kakeya estimate for 3-parameter families of curves satisfying stable geometric conditions that we call coniness and twistiness. The latter estimate extends ideas of Katz, Wu, and Zahl from the study of $\mathrm{SL}_2$-Kakeya sets.

math.CA↗

arXiv2026

Bourgain's condition, sticky Kakeya, and new examples

We prove that in all dimensions at least 3 and for any Hörmander-type phase function satisfying Bourgain's condition, the sticky case of the corresponding curved Kakeya conjecture reduces to the sticky case of the classical Kakeya conjecture. This supports a conjecture of Guo--Wang--Zhang that an oscillatory integral operator satisfies the same $L^p$ bounds as in the restriction conjecture exactly when its phase function satisfies Bourgain's condition. Our result follows from a new geometric characterization of Bourgain's condition in terms of straightening curved $δ$-tubes in a $δ^{1/2}$-tube. We construct examples in all dimensions at least 3 which show this local straightening property does not persist in a larger tube and, in particular, these are the first phase functions satisfying Bourgain's condition for which there is no diffeomorphism taking the corresponding families of curves to lines. This suggests that a general to sticky reduction in the spirit of Wang--Zahl needs substantial new ideas, and we take initial steps in this direction. We expect these examples to serve as a natural testing ground.

math.CA↗

arXiv2026

Curved Kakeya problems and the projective geometry of paths

We introduce a general framework for curved Kakeya problems in $\mathbb{R}^n$, encompassing those arising from Hörmander-type oscillatory integrals. Every family of curves determines a spray geometry, which allows us to use the projective geometry of paths in the study of curved Kakeya problems. We focus on the two extremes of the "best" and "worst" possible behaviors of curved Kakeya sets. We characterize when the incidence structure underlying Wolff's hairbrush argument persists. In particular, we prove that the existence of many totally geodesic surfaces, as required by Wolff's hairbrush argument, is equivalent to projective flatness of the associated spray. Within this projectively flat class, Bourgain's condition provides a clean dichotomy: when it holds, the family is direction-equivalent to a Bochner--Riesz type family of lines and satisfies the Katz--Wolff condition, and thus the Wang--Zahl result is applicable; when it fails, every totally geodesic surface supports a two-dimensional Kakeya set. We also show that under an extra semi-algebraic assumption, a family of curves in $\mathbb{R}^3$ admits a curved Kakeya set of Hausdorff dimension $2$ if and only if it admits a curved Kakeya set contained in a surface. Equivalently, if this compression is absent, every associated curved Kakeya set has dimension strictly greater than $2$.

math.CA↗

arXiv2024

Gaps, Ambiguity, and Establishing Complexity-Class Containments via Iterative Constant-Setting

Cai and Hemachandra used iterative constant-setting to prove that Few $\subseteq$ $\oplus$P (and thus that FewP $\subseteq$ $\oplus$P). In this paper, we note that there is a tension between the nondeterministic ambiguity of the class one is seeking to capture, and the density (or, to be more precise, the needed "nongappy"-ness) of the easy-to-find "targets" used in iterative constant-setting. In particular, we show that even less restrictive gap-size upper bounds regarding the targets allow one to capture ambiguity-limited classes. Through a flexible, metatheorem-based approach, we do so for a wide range of classes including the logarithmic-ambiguity version of Valiant's unambiguous nondeterminism class UP. Our work lowers the bar for what advances regarding the existence of infinite, P-printable sets of primes would suffice to show that restricted counting classes based on the primes have the power to accept superconstant-ambiguity analogues of UP. As an application of our work, we prove that the Lenstra-Pomerance-Wagstaff Conjecture implies that all (O(1) + loglogn)-ambiguity NP sets are in the restricted counting class $\rm RC_{PRIMES}$.

cs.CC↗

arXiv2024

A Study Guide for "A Restriction Estimate using Polynomial Partitioning"

This manuscript is intended as an accompaniment to Guth's "A restriction estimate using polynomial partitioning". We begin by summarizing the core ideas of the proof, elaborating the history and development of the techniques therein. From there, we provide supplementary details on some of the standard methods and more technical arguments which may be unfamiliar or less accessible to readers not yet acquainted with the paper. We also provide a summary of some more recent developments since the publication of Guth's work.

math.CA↗

arXiv2023

Notions of Tensor Rank

Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the "complexity" of these forms, and are thus also important. While there is one single definition of rank that completely captures the complexity of matrices (and thus linear transformations), there is no definitive analog for tensors. Rather, many notions of tensor rank have been defined over the years, each with their own set of uses. In this paper we survey the popular notions of tensor rank. We give a brief history of their introduction, motivating their existence, and discuss some of their applications in computer science. We also give proof sketches of recent results by Lovett, and Cohen and Moshkovitz, which prove asymptotic equivalence between three key notions of tensor rank over finite fields with at least three elements.

cs.CC↗

arXiv2022

Trees of Dot Products in Thin Subsets of $\mathbb R^d$

A. Iosevich and K. Taylor showed that compact subsets of $\mathbb R^d$ with Hausdorff dimension greater than $(d+1)/2$ contain trees with gaps in an open interval. Under the same dimensional threshold, we prove the analogous result where distance is replaced by the dot product. We additionally show that the gaps of embedded trees of dot products are prevalent in a set of positive Lebesgue measure, and for Ahlfors-David regular sets, the number of trees with given gaps agrees with the regular value theorem.

math.CA↗

arXiv2021

On Salum's Algorithm for $\mathrm{X3SAT}$

This is a commentary on, and critique of, Latif Salum's paper titled "Tractability of One-in-three $\mathrm{3SAT}$: $\mathrm{P} = \mathrm{NP}$." Salum purports to give a polynomial-time algorithm that solves the $\mathrm{NP}$-complete problem $\mathrm{X3SAT}$, thereby claiming $\mathrm{P} = \mathrm{NP}$. The algorithm, in short, fixes the polarity of a variable, carries out simplifications over the resulting formula to decide whether to keep the value assigned or flip the polarity, and repeats with the remaining variables. One thing this algorithm does not do is backtrack. We give an illustrative counterexample showing why the lack of backtracking makes this algorithm flawed.

cs.CC↗