arXiv subjects

Will Burstein

Publications and source records attributed to Will Burstein.

8 recordsLinked to original sources

arXiv2026

A Hierarchical Online-Compression Approach to Proportional Subsets of Bounded Orthonormal Systems

Let $(ϕ_i)_{i=1}^n$ be an orthonormal system on a probability space with $\|ϕ_i\|_\infty\leq L$. We prove that, for every $2\leq k<n$, there is a set $I$ of at least $n-k$ indices such that $$ \|f\|_2\leq CM_k\sqrt{\log(e+M_k)}\|f\|_1 $$ for every $f$ in the span of $(ϕ_i)_{i\in I}$, where $M_k=L\sqrt{\frac{n}{k}\log(2k)}$. This replaces the logarithmic exponent $\frac{5}{2}$ in the corresponding theorem of Guédon, Mendelson, Pajor, and Tomczak-Jaegermann by $\frac{1}{2}$. We also obtain a partition into two sets of cardinality $\frac{n}{2}+O(\sqrt n)$ for which both spans satisfy the resulting norm comparison, and for uniformly bounded systems the proportional estimate is $C\sqrt{\log n\log\log n}$ for sufficiently large $n$. This is the third paper in a series systematically using gradient descent as a proof mechanism in harmonic analysis, following the $Λ(p)$ and Bernoulli restricted isometry papers of Burstein, Iosevich, and Krause, and the descent here is entropic mirror descent. We represent each function in an $L_1$ ball by a probability measure of controlled relative entropy, and this entropy pays for the corrections to the predictions of its coefficients. Processing the thresholds from large to small, we obtain a single weighted bound for all the corrections, which we turn into a finite description of approximate coefficient squares and then into a conditional Rademacher estimate, and symmetrization completes the argument directly on the $L_1$ ball. We also record consequences for concentration and for recovery from coefficient measurements.

math.CA↗

arXiv2026

Fourier Ratio and Restricted Isometry via Entropic Descent

In this second paper on descent methods as proof mechanisms in harmonic analysis (the first treated Bourgain's $Λ(p)$ selection theorem), entropic mirror descent gives an improved restricted isometry bound, improving sampling and recovery bounds in the Fourier Ratio program. Let $H$ be a complex $n\times n$ matrix with $H^*H=nI$ and $|H_{ij}|=1$. For fixed $0<\varepsilon<\frac14$ and $A_0>0$, independent Bernoulli row selection with expected cardinality $C_{\varepsilon,A_0}r\log\frac{2en}{r}\log(2r)\leq m\leq n/2$ preserves, after normalization by $\sqrt m$, the squared norm of every complex vector $y$ with $\|y\|_1\leq\sqrt r\|y\|_2$ within a factor $1\pm\varepsilon$, with failure probability at most $Cn^{-A_0}$. This includes every $r$-sparse vector and also fully supported ones; support size enters only through this inequality. For $y=\widehat f$ the condition is exactly $\operatorname{FR}(f)^2\leq r$, where $\operatorname{FR}(f)=\|\widehat f\|_1/\|\widehat f\|_2$, giving uniform energy sampling and approximate recovery even for signals with full Fourier support. The same holds for a uniform subset of prescribed cardinality, and for discrete Fourier and real Hadamard matrices. At fixed accuracy the bound removes one sparsity logarithm from the earlier count and replaces $\log n$ by $\log\frac{2en}{r}$; for Walsh matrices it matches, up to constants, the lower bound of Błasiok et al.\ in their range. One relative-entropy potential controls the corrections of an amplitude predictor at every scale; counting them on the symmetric difference of two samples gives the uniform estimate. We also prove the sufficient bound $C_{A_0}\varepsilon^{-5}r\log\frac{2en}{r}\log\frac{2r}{\varepsilon}$, and stable sparse recovery from $Cs\log\frac{2en}{s}\log(2s)$ measurements, with error controlled by the noise and the best $s$-term approximation.

cs.IT↗

arXiv2026

Pointwise Convergence of Random Ergodic Averages at the $L^1$ Endpoint

In his highly influential paper, \emph{On the maximal ergodic theorem for certain subsets of the integers}, Bourgain introduced the study of pointwise convergence of ergodic averages along randomly generated subsets of the integers, developing a robust $L^p$-theory for $p>1$. More recently, this theory was partially complemented at the $L^1$ endpoint by work of LaVictoire, who treated averages along random sequences that are ``slightly denser" than the set of squares, for which universal $L^1$ pointwise convergence is known to fail. In this work, we prove that, almost surely, for every measure-preserving system and every integrable function, the ergodic averages formed by sampling along random sequences with quadratic growth converge almost everywhere. Our proof combines harmonic-analytic methods developed by M.~Christ with a ``random sampling" of quadratic forms, thereby overcoming the main uniformity obstruction.

math.DS↗

arXiv2026

Bourgain's Lambda(p) selection theorem: a greedy proof with polynomial failure bounds

We give a self-contained proof of Bourgain's finite $Λ(p)$ selection theorem with an explicit probability bound. For every $p>2$ and $N\ge2$, given $N$ pairwise orthogonal functions bounded by one on a probability space, a uniformly chosen subsystem of cardinality $\lceil N^{\frac{2}{p}}\rceil$ satisfies the $Λ(p)$ inequality with probability at least $1-\frac{1}{N}$, with a constant depending only on $p$. The inequality holds simultaneously for all complex coefficient vectors. More generally, for every fixed $A>0$, the success probability is at least $1-\frac{1}{N^A}$ with a constant depending only on $p$ and $A$, and independent of $N$. The proof uses a greedy approximation in $L^r$, with $r>2$, whose potential decreases throughout all approximation scales. A single weighted bound on the cumulative number of updates controls a union bound over discrete update histories, and increasing the weight assigned to each history gives the prescribed failure exponent without changing the cardinality exponent.

math.CA↗

arXiv2026

The Fourier Ratio: A Unifying Measure of Complexity for Recovery, Localization, and Learning

We introduce a generalized Fourier ratio, the $\ell^1/\ell^2$ norm ratio of coefficients in an \emph{arbitrary} orthonormal system, as a single, basis-invariant measure of \emph{effective dimension} that governs fundamental limits across signal recovery, localization, and learning. First, we prove that functions with small Fourier ratio can be stably recovered from random missing samples via $\ell^1$ minimization, extending and clarifying compressed sensing guarantees for general bounded orthonormal systems. Second, we establish a sharp \emph{localization obstruction}: any attempt to localize recovery to subslices of a product space necessarily inflates the Fourier ratio by a factor scaling with the square root of the slice count, demonstrating that global complexity cannot be distributed locally. Finally, we show that the same parameter controls key complexity-theoretic measures: it provides explicit upper bounds on Kolmogorov rate-distortion description length and on the statistical query (SQ) dimension of the associated function class. These results unify analytic, algorithmic, and learning-theoretic constraints under a single complexity parameter, revealing the Fourier ratio as a fundamental invariant in information-theoretic signal processing.

math.CA↗

arXiv2025

Fourier minimization and imputation of time series

One of the most common procedures in modern data analytics is filling in missing values in times series. For a variety of reasons, the data provided by clients to obtain a forecast, or other forms of data analysis, may have missing values, and those values need to be filled in before the data set can be properly analyzed. Many freely available forecasting software packages, such as the sktime library, have built-in mechanisms for filling in missing values. The purpose of this paper is to adapt the classical $L^1$ minimization method for signal recovery to the filling of missing values in times. The theoretical justifications of these methods leverage results by Bourgain (\cite{Bourgain89}), Talagrand (\cite{Talagrand98}), the second and the third listed authors (\cite{IM24}), and the result by the second listed author, Kashin, Limonova and the third listed author (\cite{IKLM24}). Brief numerical tests for these algorithms are given but more extensive will be discussed in a companion paper.

math.CA↗

arXiv2025

$Λ_p$ Style Bounds in Orlicz Spaces Close to $L^2$

Let $(φ_i)_{i=1}^n$ be mutually orthogonal functions on a probability space such that $\|φ_i\|_\infty \leq 1 $ for all $i \in [n]$. Let $α> 0$. Let $Φ(u) = u^2 \log^α(u)$ for $u \geq u_{0}$, and $Φ(u) = c(α) u^2$ otherwise. $u_0 \geq e$ and $c(α)$ are constants chosen so that $Φ$ is a Young function, depending only on $α$. Our main result shows that with probability at least $1/4$ over subsets $I$ of $[n]$, where $I$ is constructed by choosing each index of $[n]$ independently from a Bernoulli distribution, the following holds: $|I| \geq \frac{n}{e \log^{α+1}(n)} $ and for any $a \in \mathbb{C}^n$, $$ \left \|\sum_{i \in I} a_i φ_i \right \|_Φ \leq K(α) \log^{\fracα{2}}(\log n) \cdot \|a\|_2. $$ $K(α)$ is a constant depending only on $α$. In the main Theorem of \cite{Ryou22}, Ryou proved the result above to a constant factor, depending on $p$ and $α$, when the Orlicz space is a $L^p(\log L)^{pα}$ space for $p > 2$ where $|I| \sim \frac{n^{2/p}}{\log^{2 α/p}(n)}$. However, their work did not extend to the case where $p=2$, an open question in \cite{Iosevich25}. Our result resolves the latter question up to $\log \log n$ factors. Moreover, our result sharpens the constants of Limonova's main result in \cite{Limonova23} from a factor of $\log n$ to a factor of $\log \log n$, if the orthogonal functions are bounded by a constant. In addition, our proof is much shorter and simpler than the latter's. Finally, to complement our main result, we give a probabilistic lower bound (subsets of $[n]$ are selected by a Bernoulli distribution over $[n]$'s indices) that matches our main result's upper bound.

math.CA↗

arXiv2024

Group Action Approaches in Erdos Quotient Set Problem

Let $\mathbb{F}_q$ denote the finite field of $q$ elements. For $E \subset \mathbb{F}_q^d$, denote the distance set $Δ(E)= \{\|x-y\|^2:=(x_1-y_1)^2+ \cdots + (x_d-y_d)^2 : (x,y)\in E^2 \}$. The Erdos quotient set problem was introduced in \cite{Iosevich_2019} where it was shown that for even $d\geq2$ that if $|E| \subset \mathbb{F}_q^2$ such that $|E| >> q^{d/2}$, then $\frac{Δ(E)}{Δ(E)}:= \{\frac{s}{t}:s,t \in Δ(E), t\not=0\} =\mathbb{F}_q^d$. The proof of the latter result is quite sophisticated and in \cite{pham2023group}, a simple proof using a group-action approach was obtained for the case of $q \equiv 3 \mod 4$ when $d=2$. In the $q \equiv 3 \mod 4$ setting, for each $r \in (\mathbb{F}_q)^2$, \cite{pham2023group} showed if $E \subset \mathbb{F}_q$, then $V(r):= \# \left\{ (a,b,c,d) \in E^2: \frac{\|a-b\|^2}{\|c-d\|^2} = r \right\} >> \frac{|E|^4}{q}$. In this work we use group action techniques in the $q \equiv 3 \mod 4$ setting, for $d=2$ and improve the results of \cite{pham2023group} by removing the assumption on $r \in (\mathbb{F}_q)^2$. Specifically we show if $d=2$ and $q \equiv 3 \mod 4$, then for each $r \in \mathbb{F}_q^*$,$V(r)\geq \frac{|E|^4}{2q}$if $|E|\geq \sqrt{2}q$ for all $r \in \mathbb{F}_q$. Finally, we improve the main result of \cite{bhowmik2023near} using our proof techniques from our quotient set results.

math.CO↗