arXiv subjects

Ben Krause

Publications and source records attributed to Ben Krause.

At least 19 recordsLinked to original sources

arXiv2026

Endpoint Estimates for Stein's Purely Quadratic Carleson Operator

We study the near $L^1$ behavior of the maximally quadratically modulated Hilbert transform \[ \mathcal{C}_2f(x) := \sup_{\lambda} \left|\operatorname{p.v.} \int_{\mathbb{R}}f(x-y)e^{2 \pi i \lambda y^2} \frac{\mathrm{d} y }{y} \right| \] and its lacunary counterpart obtained by restricting $\lambda$ to $2^{\mathbb{Z}}.$ We prove that if $\Phi$ is a Young function satisfying $ \Phi(t)=o\bigl(t\log_2t\bigr)$ then $\mathcal{C}_{2,\mathsf{lac}}$, and therefore $\mathcal{C}_2$ as well, does not satisfy a corresponding $\Phi$-modular estimate. In particular, neither the lacunary nor the full quadratic operator is of weak type $(1,1)$. In the positive direction, we establish an $L\log_1 L$ modular estimate for $\mathcal{C}_2$ and a $L (\log_2L)^2 \log_4L$ estimate for $\mathcal{C}_{2,\mathsf{lac}}.$

math.CA↗

arXiv2026

Quantitative bounds for sets lacking polynomial progressions with shifted prime difference

We prove quantitative polynomial Szemer\'edi-type theorems involving polynomial progressions with shift parameter restricted to the set of shifted primes $\mathbb{P}-1$. The types of configurations covered are distinct degree progressions and progressions involving integer multiples of a fixed polynomial. For nonlinear configurations of length at least three, these results provide the first quantitative versions of such theorems. In the linear case, our results improve on work by the last two authors. Our density bounds are strongest in the case of distinct degree polynomials, where they give polylogarithmic bounds, of the same shape as recent bounds by Shao and Wang with integer shifts. The proofs combine recent quantitative results for polynomial configurations in the integers with quantitative Gowers uniformity bounds of the primes. For multiples of a fixed polynomial, we adapt a comparison argument of Altman and Sawhney to obtain uniformity over the polynomial families produced by the $W$-trick. For distinct degree progressions, we establish a comparison between prime-weighted and unweighted polynomial counts that is uniform throughout the density increment argument and accounts for a possible Siegel zero.

math.NT↗

arXiv2026

Pointwise Convergence of Ergodic Averages Along Integer Cantor Sets

Let $d \geq 3$, \[ D \subsetneq \{ 0,1,\dots,d-1\}, \qquad |D| \geq 2, \ 0 \in D \] be a finite alphabet, and define the integer Cantor set \begin{align} \mathcal{C} := \mathcal{C}_{D} := \bigcup_{J \geq 0} \Big\{ \sum_{j =0}^J a_j d^j : a_j \in D \Big\}. \end{align} We prove that for any $\sigma$-finite measure-preserving system, $(X,\mu,T)$, and any $f \in L^p(X)$, $2\leq p<\infty$, the ergodic averages \begin{align} \frac{1}{|\mathcal{C}_N|} \sum_{n \in \mathcal{C}_N } f(T^n x), \qquad \mathcal{C}_N := \mathcal{C} \cap \{1,2,\dots,N \} \end{align} converge $\mu$-almost everywhere. By rescaling, this allows us to resolve the question of lacunary differentiation of Cantor measures at self-similar scales: if \begin{align} \mathcal{C}' := \Big\{ \sum_{j \geq 1} a_j d^{-j} : a_j \in D \Big\} \subset [0,1] \end{align} is a real-variable Cantor set, and $\nu$ denotes its natural measure, then we prove that \begin{align} \lim_{k \to \infty} \int f(x-d^{-k} t) \ d\nu(t) = f(x) \end{align} Lebesgue almost-everywhere for any $f \in L^2_{\text{loc}}(\mathbb{R})$.

math.DS↗

arXiv2026

A Variation Norm Carleson Theorem Along the Primes

Let $\Lambda$ denote the von Mangoldt function; we prove that for each $r > 2$, there exist constants \[ r' < \mathbf{c}(r) < 2 < \mathbf{C}(r), \qquad \lim_{r \to \infty} \mathbf{c}(r) = 1, \ \lim_{r \to \infty} \mathbf{C}(r) = \infty \] so that the discrete variational Carleson operator along the primes \begin{align} \mathcal{V}^r \Big( \sum_{n \neq 0} f(x-n) \Lambda(|n|) \frac{e^{2\pi i \lambda n}}{n} : \lambda \in \mathbb{T} \Big) \end{align} is bounded on $\ell^p$ for all $\mathbf{c}(r) < p < \mathbf{C}(r)$, while the variation is unbounded when $p \leq r'$. At the non-variational endpoint, the same argument gives the sharp maximal result: the prime Carleson operator \[ \sup_{\lambda\in\mathbb T} \Big|\sum_{n\neq0} f(x-n)\Lambda(|n|)\frac{e^{2\pi i\lambda n}}{n}\Big| \] is bounded on $\ell^p(\mathbb Z)$ for the full expected range $1<p<\infty$. The proof gives a new mechanism for treating modulation-invariant singular integrals after arithmetic sparsification. It combines higher-order Fourier uniformity, a variable-coefficient multi-frequency principle in the spirit of Bourgain, and an additive-combinatorial inverse argument. A key step is a reduction to finite periodic models, where the Ramanujan structure of the major arcs is converted into a sharp estimate for structured atoms by elementary number theory.

math.CA↗

arXiv2026

Double Recurrence and Almost Sure Convergence: Primes and Weighted Theory

Let $(X,\mu)$ be a probability space equipped with an invertible, measure-preserving transformation $T\colon X \to X$. We exhibit a wide class of weights $w$ so that whenever $f,g \in L^{\infty}(X)$, the bilinear ergodic averages \[ \frac{1}{N} \sum_{n \leq N} w(n)\, T^{an}f \cdot T^{bn}g, \qquad a,b \in \mathbb{Z} \] converge $\mu$-almost surely. This class encompasses the von Mangoldt function, resolving Problem 12 from Frantzikinakis' survey on open problems in ergodic theory, the divisor function, the sum-of-two-squares representation function, etc., as well as their restrictions to lower-density Piatetski-Shapiro sequences of the form $\{\lfloor k^{c}\rfloor : k \in \mathbb{N}\}$, $1 \leq c < 7/6$. Our methods combine combinatorial number theory and higher-order Fourier analysis with classical Fourier-analytic/martingale-based methods; the role of $U^{3}$ analysis is particularly significant.

math.DS↗

arXiv2026

The Wiener Wintner Theorem Along the Primes

We prove the following Wiener-Wintner Theorem along the sequence of prime times, the first extension of the Wiener-Wintner Theorem to arithmetic sequences: for every probability space, $(X, \nu),$ equipped with a measure-preserving transformation, $T : X \to X,$ and every $f \in L^p(X), 1 < p \leq \infty$, there exists a set of full probability, $X_f \subset X$ with $\nu(X_f) = 1,$ so that for all $\omega \in X_f$, \[ \frac{1}{N} \sum_{n \leq N} e^{ 2 \pi i p_n \theta} f(T^{p_n} \omega) \] converges for all $\theta \in [0,1]$; above, $\{2 = p_1 < p_2 < \dots\}$ are an enumeration of the primes. Our proof lives at the interface of classical Fourier analysis, combinatorial number theory, higher order Fourier analysis, and pointwise ergodic theory, with U^3 theory playing an important role; our $U^3$-estimates for Heath-Brown models of the von Mangoldt function may be of independent interest.

math.DS↗

arXiv2025

Cross representations of additive complements of $r$-th powers

Let $\mathbb{N}$ be the set of natural numbers and $\mathcal{S}_r=\big\{1^r, 2^r, 3^r,\cdots\big\}$ the set of $r$-th powers, where $r\ge 2$ is a natural number. Let $\mathcal{W}_r$ be an additive complement of $\mathcal{S}_r$ and $$ f_r(n)=\#\big\{(w,m^r)\in \mathcal{W}_r\times \mathcal{S}_r: n=w+m^r\big\}. $$ Motivated by a 1993 conjecture of Cilleruelo, we show that $$ \sum_{n\le N}f_r(n)-N\gg_r N^{1-\frac{1}{r}}. $$ Previously, the bound was only proved for $r=2$. In the case $r=2$, the lower bound above can be made more explicit as $$ \sum_{n\le N}f_2(n)-N\gg N^{3/4-o(1)}, $$ which improves the previous bound $N^{1/2}$ due to Ding, Sun, Wang and Xia.

math.NT↗

arXiv2025

A Hard-Analytic Proof of "Most" Polynomial Wiener-Wintner Theorems for Infinite Measure Spaces

We provide a new proof of ``most" cases of the polynomial Wiener-Wintner theorem for $\sigma$-finite spaces, using hard-analytic methods. Specifically, we prove that whenever $(X,\mu,T)$ is a $\sigma$-finite measure-preserving system, and $f \in L^p(X), \ 1 \leq p < \infty$, there exists a co-null set $X_f \subset X$ so that for all $\omega \in X_f$ \[ \frac{1}{N} \sum_{n \leq N} e^{2 \pi i P(n)} f(T^n \omega) \] converges for all polynomials $P$ which are either linear, or vanish to degree $2$ at the origin.

math.DS↗

arXiv2025

On Multi-linear Maximal Operators Along Homogeneous Curves

Suppose that \[ \vec{\gamma}(t) := (\gamma_1(t),\dots,\gamma_n(t)) = (a_1 t^{d_1},\dots,a_n t^{d_n}), \; \; \; 1\leq d_1 < \dots < d_n, \ a_i \neq 0\] is a homogeneous polynomial curve. We prove that whenever $p_1,\dots,p_n > 1$ and $\frac{1}{p} = \sum_{j=1}^n \frac{1}{p_j} \leq 1$, there exists an absolute constant $0 < C = C_{p_1,\dots,p_n;\vec{\gamma}} < \infty$ so that \[ \| \sup_{r > 0} \ \frac{1}{r} \int_{0}^r \prod_{i=1}^n |f_i(x-\gamma_i(t))| \ dt \|_{L^p(\mathbb{R})} \leq C \cdot \prod_{i=1}^n \| f_j \|_{L^{p_j}(\mathbb{R})}. \] Our main tool is a smoothing estimate, adapted from work of Kosz-Mirek-Peluse-Wright.

math.CA↗

arXiv2025

On Polynomial Progressions Inside Sets of Large Dimension

In this note we connect Sobolev estimates in the context of polynomial averages e.g. \[ \| \int_0^1 \prod_{k=1}^m f_k(x-t^k) \|_{1} \leq \text{Const} \cdot 2^{-\text{const} \cdot l} \prod_{i=1}^m \| f_k \|_m \] whenever some $f_i$ vanishes on $\{ |\xi| \leq 2^l \}$ to the existence of polynomial progressions inside of sets of sufficiently large Hausdorff dimension, in analogy with work of Peluse in the discrete context. Our strongest (unconditional) result builds off deep work of Hu-Lie and is as follows: suppose that $\mathcal{P} = \{P_1,P_2,P_3\}$ vanish at the origin at different rates, and that $E \subset [0,1]$ has sufficiently large Hausdorff dimension, \[ 1 - \text{const}(\mathcal{P}) < \text{dim}_H(E) < 1 \] and Hausdorff content bounded away from zero, sufficiently large in terms of its dimension. Then $E$ contains a non-trivial polynomial progression of the form \[ \{ x , x - P_1(t), x - P_2(t), x - P_3(t) \} \subset E, \; \; \; t \neq 0. \] We also provide a short proof that whenever $E$ has sufficiently large Hausdorff dimension and Fourier dimension $> 1/2$, it necessarily contains a non-trivial generalized three-term arithmetic progression of the form \[ \{ x, x - \theta_1 t, x- \theta_2 t\} \subset E, \; \; \; \theta_i \in \mathbb{Q},\ t \neq 0.\]

math.CA↗

arXiv2025

Quantitative Convergence for Sparse Ergodic Averages in $L^1$

We provide a unified framework to proving pointwise convergence of sparse sequences, deterministic and random, at the $L^1(X)$ endpoint. Specifically, suppose that \[ a_n \in \{ \lfloor n^c \rfloor, \min\{ k : \sum_{j \leq k} X_j = n\} \} \] where $X_j$ are Bernoulli random variables with expectations $\mathbb{E} X_j = n^{-\alpha}$, and we restrict to $1 < c < 7/6, \ 0 < \alpha < 1/2$. Then (almost surely) for any measure-preserving system, $(X,\mu,T)$, and any $f \in L^1(X)$, the ergodic averages \[ \frac{1}{N} \sum_{n \leq N} T^{a_n} f \] converge $\mu$-a.e. Moreover, our proof gives new quantitative estimates on the rate of convergence, using jump-counting/variation/oscillation technology, pioneered by Bourgain. This improves on previous work of Urban-Zienkiewicz, and Mirek, who established the above with $c = \frac{1001}{1000}, \ \frac{30}{29}$, respectively, and LaVictoire, who established the random result, all in a non-quantitative setting.

math.DS↗

arXiv2025

An Approach To Endpoint Problems in Oscillatory Singular Integrals

In this note we provide a quick proof that maximal truncations of oscillatory singular integrals are bounded from $L^1(\mathbb{R})$ to $L^{1,\infty}(\mathbb{R})$. The methods we use are entirely elementary, and rely only on pigeonholing and stationary phase considerations.

math.CA↗

arXiv2025

Multi-Frequency Oscillation Estimates Arising in Pointwise Ergodic Theory

We prove essentially optimal $L^p(\mathbb{R})$-estimates for variational variants of the maximal Fourier multiplier operators considered by Bourgain in his work on pointwise convergence of polynomial ergodic averages. As a corollary of our methods, we are able to quickly extend a result of Bourgain, namely the pointwise convergence of ergodic averages of integer parts of real-variables polynomials, to a broader class of functions, previously considered in a wide range of contexts by Boshernitzan-Jones-Wierdl. Namely, the following averages converge almost everywhere \[ \frac{1}{N} \sum_{n \leq N} T^{\lfloor P(n) \rfloor} f, \; \; \; f \in L^p(X,\mu), \ P \in \mathbb{R}[\cdot], \] for any $\sigma$-finite measure space equipped with a measure-preserving transformation, $T:X \to X$, whenever $1 < p \leq \infty$ if $P$ is linear, and $4/3 < p \leq \infty$ otherwise.

math.CA↗

arXiv2025

A Unified Approach to Two Pointwise Ergodic Theorems: Double Recurrence and Return Times

We present a unified approach to extensions of Bourgain's Double Recurrence Theorem and Bourgain's Return Times Theorem to integer parts of the Kronecker sequence, emphasizing stopping times and metric entropy. Specifically, we prove the following two results for each $\alpha \in \mathbb{R}$: First, for each $\sigma$-finite measure-preserving system, $(X,\mu,T)$, and each $f,g \in L^{\infty}(X)$, for each $\gamma \in \mathbb{Q}$ the bilinear ergodic averages \[ \frac{1}{N} \sum_{n \leq N} T^{\lfloor \alpha n \rfloor } f \cdot T^{\lfloor \gamma n \rfloor} g \] converge $\mu$-a.e.; Second, for each aperiodic and countably generated measure-preserving system, $(Y,\nu,S)$, and each $g \in L^{\infty}(Y)$, there exists a subset $Y_{g} \subset Y$ with $\nu(Y_{g})= 1$ so that for all $\gamma \in \mathbb{Q}$ and $\omega \in Y_{g}$, for any auxiliary $\sigma$-finite measure-preserving system $(X,\mu,T)$, and any $f \in L^{\infty}(X)$, the ``return-times" averages \[ \frac{1}{N} \sum_{n \leq N} T^{\lfloor \alpha n \rfloor} f \cdot S^{\lfloor \gamma n \rfloor } g(\omega) \] converge $\mu$-a.e. Moreover, in both cases the sets of convergence are identical for all $\gamma \in \mathbb{Q}$.

math.DS↗

arXiv2024

Pointwise convergence of bilinear polynomial averages over the primes

We show that on a $\sigma$-finite measure preserving system $X = (X,\nu, T)$, the non-conventional ergodic averages $$ \mathbb{E}_{n \in [N]} \Lambda(n) f(T^n x) g(T^{P(n)} x)$$ converge pointwise almost everywhere for $f \in L^{p_1}(X)$, $g \in L^{p_2}(X)$, and $1/p_1 + 1/p_2 \leq 1$, where $P$ is a polynomial with integer coefficients of degree at least $2$. This had previously been established with the von Mangoldt weight $\Lambda$ replaced by the constant weight $1$ by the first and third authors with Mirek, and by the M\"obius weight $\mu$ by the fourth author. The proof is based on combining tools from both of these papers, together with several Gowers norm and polynomial averaging operator estimates on approximants to the von Mangoldt function of ''Cram\'er'' and ''Heath-Brown'' type.

math.DS↗

arXiv2024

A Framework for Evaluating Appropriateness, Trustworthiness, and Safety in Mental Wellness AI Chatbots

Large language model (LLM) chatbots are susceptible to biases and hallucinations, but current evaluations of mental wellness technologies lack comprehensive case studies to evaluate their practical applications. Here, we address this gap by introducing the MHealth-EVAL framework, a new role-play based interactive evaluation method designed specifically for evaluating the appropriateness, trustworthiness, and safety of mental wellness chatbots. We also introduce Psyfy, a new chatbot leveraging LLMs to facilitate transdiagnostic Cognitive Behavioral Therapy (CBT). We demonstrate the MHealth-EVAL framework's utility through a comparative study of two versions of Psyfy against standard baseline chatbots. Our results showed that Psyfy chatbots outperformed the baseline chatbots in delivering appropriate responses, engaging users, and avoiding untrustworthy responses. However, both Psyfy and the baseline chatbots exhibited some limitations, such as providing predominantly US-centric resources. While Psyfy chatbots were able to identify most unsafe situations and avoid giving unsafe responses, they sometimes struggled to recognize subtle harmful intentions when prompted in role play scenarios. Our study demonstrates a practical application of the MHealth-EVAL framework and showcases Psyfy's utility in harnessing LLMs to enhance user engagement and provide flexible and appropriate responses aligned with an evidence-based CBT approach.

cs.HC↗

arXiv2024

AutoGRAMS: Autonomous Graphical Agent Modeling Software

We introduce the AutoGRAMS framework for programming multi-step interactions with language models. AutoGRAMS represents AI agents as a graph, where each node can execute either a language modeling instruction or traditional code. Likewise, transitions in the graph can be governed by either language modeling decisions or traditional branch logic. AutoGRAMS supports using variables as memory and allows nodes to call other AutoGRAMS graphs as functions. We show how AutoGRAMS can be used to design highly sophisticated agents, including self-referential agents that can modify their own graph. AutoGRAMS's graph-centric approach aids interpretability, controllability, and safety during the design, development, and deployment of AI agents. We provide our framework as open source at https://github.com/autograms/autograms .

cs.CL↗

arXiv2024

Averages over the Gaussian Primes: Goldbach's Conjecture and Improving Estimates

We prove versions of Goldbach conjectures for Gaussian primes in arbitrary sectors. Fix an interval $ω\subset \mathbb{T}$. There is an integer $N_ω$, so that every odd integer $n$ with $N(n)>N_ω$ and $\text{dist}( \text{arg}(n) , \mathbb{T}\setminus ω) > (\log N(n)) ^{-B}$, is a sum of three Gaussian primes $n=p_1+p_2+p_3$, with $\text{arg}(p_j) \in ω$, for $j=1,2,3$. A density version of the binary Goldbach conjecture in a sector is also proved.

math.NT↗