arXiv subjects

James Leng

Publications and source records attributed to James Leng.

17 recordsLinked to original sources

arXiv2025

Quantitative pyjama

The "pyjama stripe" with parameter $\varepsilon>0$ is the set $E(\varepsilon)$ of all complex numbers $z$ such that the distance from $\Re(z)$ to the nearest integer is at most $\varepsilon$. The Pyjama Problem of Iosevich, Kolountzakis, and Matolcsi asks whether, for every choice of $\varepsilon>0$, it is possible to cover the entire complex plane with finitely many rotations of $E(\varepsilon)$ around the origin. Manners obtained an affirmative answer to this question by studying a $\times 2, \times 3$-type problem over a suitable solenoid. Manners's argument provided no quantitative bounds (in terms of $\varepsilon$) on the number of rotations required, and Green has highlighted the problem of obtaining such quantitative bounds. Our main result is that $\exp\exp\exp(\varepsilon^{-O(1)})$ rotations of $E(\varepsilon)$ suffice to cover the complex plane. Our analysis makes use of the entropic tools developed by Bourgain, Lindenstrauss, Michel, and Venkatesh for quantitative $\times 2, \times 3$-type results.

math.DS↗

arXiv2025

Structured extensions and multi-correlation sequences

We show that every multi-correlation sequence is the sum of a generalized nilsequence and a null-sequence. This proves a conjecture of N. Frantzikinakis. A key ingredient is the reduction of ergodic multidimensional inverse theorems to analogous finitary inverse theorems, offering a new approach to the structure theory of multidimensional Host-Kra factors. This reduction is proven by combining the methods of Tao (2015) with the Furstenberg correspondence principle. We also prove the analogous multidimensional finitary inverse theorem with quasi-polynomial bounds.

math.DS↗

arXiv2024

Vinogradov's theorem for primes with restricted digits

Let $g$ be sufficiently large, $b\in\{0,\ldots,g-1\}$, and $\mathcal{S}_b$ be the set of integers with no digit equal to $b$ in their base $g$ expansion. We prove that every sufficiently large odd integer $N$ can be written as $p_1 + p_2 + p_3$ where $p_i$ are prime and $p_i\in \mathcal{S}_b$.

math.NT↗

arXiv2024

Quantitative concatenation for polynomial box norms

Using PET and quantitative concatenation techniques, we establish box-norm control with the "expected" directions for counting operators for general multidimensional polynomial progressions, with at most polynomial losses in the parameters. Such results are often useful first steps towards obtaining explicit upper bounds on sets lacking instances of given such progressions. In the companion paper arXiv:2407.08637, we complete this program for sets in $[N]^2$ lacking nondegenerate progressions of the form $(x, y), (x + P(z), y), (x, y + P(z))$, where $P \in \mathbb{Z}[z]$ is any fixed polynomial with an integer root of multiplicity $1$.

math.CO↗

arXiv2024

Corners with polynomial side length

A $P$-polynomial corner, for $P \in \mathbb{Z}[z]$ a polynomial, is a triple of points $(x,y),\; (x+P(z),y),\; (x,y+P(z))$ for $x,y,z \in \mathbb{Z}$. In the case where $P$ has an integer root of multiplicity $1$, we show that if $A \subseteq [N]^2$ does not contain any nontrivial $P$-polynomial corners, then $$|A| \ll_P \frac{N^2}{(\log\log\log N)^c}$$ for some absolute constant $c>0$. This simultaneously generalizes a result of Shkredov about corner-free sets and a recent result of Peluse, Sah, and Sawhney about sets without $3$-term arithmetic progressions of common difference $z^2-1$. The main ingredients in our proof are a multidimensional quantitative concatenation result from our companion paper arXiv:2407.08636 and a novel degree-lowering argument for box norms.

math.CO↗

arXiv2024

Quasipolynomial bounds on the inverse theorem for the Gowers $U^{s+1}[N]$-norm

We prove quasipolynomial bounds on the inverse theorem for the Gowers $U^{s+1}[N]$-norm. The proof is modeled after work of Green, Tao, and Ziegler and uses as a crucial input recent work of the first author regarding the equidistribution of nilsequences. In a companion paper, this result will be used to improve the bounds on Szemer\'{e}di's theorem.

math.CO↗

arXiv2024

Improved Bounds for Szemer\'{e}di's Theorem

Let $r_k(N)$ denote the size of the largest subset of $[N] = \{1,\ldots,N\}$ with no $k$-term arithmetic progression. We show that for $k\ge 5$, there exists $c_k>0$ such that \[r_k(N)\ll N\exp(-(\log\log N)^{c_k}).\] Our proof is a consequence of recent quasipolynomial bounds on the inverse theorem for the Gowers $U^k$-norm as well as the density increment strategy of Heath-Brown and Szemer\'{e}di as reformulated by Green and Tao.

math.CO↗

arXiv2023

Efficient Equidistribution of Nilsequences

We give improved bounds for the equidistribution of (multiparameter) nilsequences subject to any degree filtration. The bounds we obtain are single exponential in dimension, improving on double exponential bounds of Green and Tao. To obtain these bounds, we overcome "induction of dimension'' which is ubiquitous throughout higher order Fourier analysis. The improved equidistribution theory is a crucial ingredient in the quasi-polynomial $U^4[N]$ inverse theorem of the author and its extension to the quasi-polynomial $U^{s + 1}[N]$ inverse theorem in joint work with Sah and Sawhney. These results lead to further applications in combinatorial number theory such as bounds for linear equations in the primes which save an arbitrary power of logarithm, which match the bounds Vinogradov obtained for the odd Goldbach conjecture.

math.NT↗

arXiv2023

Improved bounds for five-term arithmetic progressions

Let $r_5(N)$ be the largest cardinality of a set in $\{1,\ldots,N\}$ which does not contain $5$ elements in arithmetic progression. Then there exists a constant $c\in (0,1)$ such that \[r_5(N)\ll \frac{N}{\exp((\log\log N)^{c})}.\] Our work is a consequence of recent improved bounds on the $U^4$-inverse theorem of the first author and the fact that $3$-step nilsequences may be approximated by locally cubic functions on shifted Bohr sets. This combined with the density increment strategy of Heath-Brown and Szemer{\'e}di, codified by Green and Tao, gives the desired result.

math.NT↗

arXiv2023

Efficient equidistribution of periodic nilsequences and applications

This is a companion paper to arXiv:2312.10772. We deduce an equidistribution theorem for periodic nilsequences and use this theorem to give two applications in arithmetic combinatorics. The first application is quasi-polynomial bounds for a certain complexity one polynomial progression, improving the iterated logarithm bound previusly obtained. The second application is a proof of the quasi-polynomial $U^4[N]$ inverse theorem. In work with Sah and Sawhney, we obtain improved bounds for sets lacking nontrivial $5$-term arithmetic progressions.

math.NT↗

arXiv2023

The fractal uncertainty principle via Dolgopyat's method in higher dimensions

We prove a fractal uncertainty principle with exponent $\frac{d}{2} - \delta + \varepsilon$, $\varepsilon > 0$, for Ahlfors--David regular subsets of $\mathbb R^d$ with dimension $\delta$ which satisfy a suitable "nonorthogonality condition". This generalizes the application of Dolgopyat's method by Dyatlov--Jin (arXiv:1702.03619) to prove the same result in the special case $d = 1$. As a corollary, we get a quantitative spectral gap for the Laplacian on convex cocompact hyperbolic manifolds of arbitrary dimension with Zariski dense fundamental groups.

math.CA↗

arXiv2022

Improved quadratic Gowers uniformity for the M\"obius function

We demonstrate that $$\|\mu\|_{U^3([N])} \ll_{A}^{\text{ineff}} \log^{-A}(N)$$ $$\|\Lambda - \Lambda_Q\|_{U^3([N])} \ll_{A}^{\text{ineff}} \log^{-A}(N)$$ for any $A > 0$ where $\Lambda_Q$ is an approximant to the von Mangoldt function and will be defined below, improving upon a bound of Tao-Ter\"av\"ainen (2021). As a consequence, among other things, we have the following: $$\mathbb{E}_{x, y \in [N], x + 3y \in [N]} \Lambda(x)\Lambda(x + y)\Lambda(x + 2y)\Lambda(x + 3y) = \mathfrak{S} + O_A(\log^{-A}(N))$$ where $\mathfrak{S}$ is the singular series for the configuration $(x, x + y, x + 2y, x + 3y)$. In fact, we show that $$\|\mu - \mu_{Siegel}\|_{U^3([N])} \ll \exp(-O(\log^{1/C}(N)))$$ $$\|\Lambda - \Lambda_{Siegel}\|_{U^3([N])} \ll \exp(-O(\log^{1/C}(N)))$$ where $\mu_{Siegel}$ and $\Lambda_{Siegel}$ are approximants of $\mu$, and $\Lambda$, respectively, representing the Siegel zero contribution of $\mu$ and are defined in the above article. To do so, we use an improvement of the $U^3$ inverse theorem due to Sanders and we follow the approach of Green and Tao (2007), opting to use the ``old-fashioned" approach to equidistribution on two-step nilmanifolds which was also considered by Green and Tao (2017), and by Gowers and Wolf (2010). To the author's knowledge, this is the first time that quadratic Fourier analysis over $\mathbb{Z}/N\mathbb{Z}$ has achieved quasi-polynomial type bounds in applications.

math.NT↗

arXiv2022

A Quantitative Bound For Szemer\'edi's Theorem for a Complexity One Polynomial Progression over $\mathbb{Z}/N\mathbb{Z}$

Let $N$ be a large prime and $P, Q \in \mathbb{Z}[x]$ two linearly independent polynomials with $P(0) = Q(0) = 0$. We show that if a subset $A$ of $\mathbb{Z}/N\mathbb{Z}$ lacks a progression of the form $(x, x + P(y), x + Q(y), x + P(y) + Q(y))$, then $$|A| \le O\left(\frac{N}{\log_{(O(1))}(N)}\right)$$ where $\log_{C}(N)$ is an iterated logarithm of order $C$ (e.g., $\log_{2}(N) = \log\log(N)$). To establish this bound, we adapt Peluse's (2018) degree lowering argument to the quadratic Fourier analysis setting to obtain quantitative bounds on the true complexity of the above progression. Our method also shows that for a large class of polynomial progressions, if one can establish polynomial-type bounds on the true complexity of those progressions, then one can establish polynomial-type bounds on Szemer\'edi's theorem for that type of polynomial progression.

math.NT↗

arXiv2020

Nonsingular transformations that are ergodic with isometric coefficients and not weakly doubly ergodic

We study two properties of nonsingular and infinite measure-preserving ergodic systems: weak double ergodicity, and ergodicity with isometric coefficients. We show that there exist infinite measure-preserving transformations that are ergodic with isometric coefficients but are not weakly doubly ergodic. We also give type $\text{III}_\lambda$ examples of such systems, $0<\lambda\leq 1$. We prove that under certain hypotheses, systems that are weakly mixing are ergodic with isometric coefficients and along the way we give an example of a uniformly rigid topological dynamical system along the sequence $(n_i)$ that is not measure theoretically rigid along $(n_i)$ for any nonsingular ergodic finite measure.

math.DS↗

arXiv2020

On finite spacer rank for words and subshifts

We define a notion of rank for words and subshifts that we call spacer rank, extending the notion of rank-one symbolic shifts of Gao and Hill. We construct infinite words of each finite spacer rank, of unbounded spacer rank, and show there exist words that do not have a spacer rank construction. We consider words that are fixed points of substitutions and give explicit conditions for the word to have an at most spacer rank two construction, and not to be rank one. We prove that finite spacer rank subshifts have topological entropy zero, and that there are zero entropy subshifts not defined by a word with a finite spacer rank construction. We also study shift systems associated with infinite words, including those associated to Sturmian sequences, which we show are spacer rank-two systems.

math.DS↗

arXiv2020

Moser's Method and Conservative Extensions of Diffeomorphisms

This paper shall be concerned with three main results. After a brief recollection of basic symplectic geometry, we prove using Moser's homotopy method a special case of the Strong Darboux Theorem found, for instance, in Theorem 21.1.6 of [Hor]. Next, we'll prove two conservative extension results for a diffeomorphism on a circle. One uses Moser's homotopy method but loses a degree of regularity. The other uses the method of generating functions as found in [BCW] and [BGV]. Finally, we'll prove a conservative extension result for a "diffeomorphism" defined on the boundary of $(0, 1)^2$ and use the techniques developed there and by [M] to prove an ambient Dacarogna-Moser Theorem.

math.CA↗

arXiv2018

Non-Rigid Rank-One Infinite Measures on the Circle

For a class of irrational numbers, depending on their Diophantine properties, we construct explicit rank-one transformations that are totally ergodic and not weakly mixing. We classify when the measure is finite or infinite. In the finite case they are isomorphic to irrational rotations. We also obtain rank-one nonrigid infinite invariant measures for irrational rotations, and, for each Krieger type, nonsingular measures on irrational rotations. In the third version, in the infinite case we use the constructions to provide examples of non-weakly mixing infinite measure-preserving ergodic transformations which do not have any nontrivial probability preserving factors with discrete spectrum, thereby answering a questions of Aaronson and Nakada and of Glasner and Weiss.

math.DS↗