arXiv subjects

Ben Doyle

Publications and source records attributed to Ben Doyle.

6 recordsLinked to original sources

arXiv2026

Subconvexity of Short $k$-Free Exponential Sums in $\mathbb{F}_q[t]$

We extend recent work of the author over $\mathbb{Z}$ into the positive characteristic setting of $\mathbb{F}_q[t]$. In particular, for a polynomial $F \in \mathbb{F}_q[t]$ of degree $N$, let $R_k(\alpha)$ denote the exponential sum over $k$-free polynomials $f$ with $\text{deg}(f-F) 0$, we prove essentially tight upper and lower bounds for the $s$-th moment of $R_k(\alpha)$ whenever $K > (\frac{1}{2}+\epsilon)N$, and in even shorter intervals when $s>1+\frac{1}{k}$. As an application of these results, we prove a lower bound of order $q^{\frac{K}{6}}$ for the $L^1$-mean of the M\"obius-twisted exponential sum over $\mathbb{F}_q[t]$ whenever $K >(\frac{1}{2}+\epsilon)N$.

math.NT↗

arXiv2026

Subconvexity of Short $k$-Free Exponential Sums

Let $S_k(\alpha;K)$ denote the exponential sum over $k$-free integers in the short interval $(N-K,N]$. For $s>0$, we prove essentially tight bounds on the $s$-th moments of $S_k(\alpha;K)$ whenever $K \gg N^{\theta_{k,s}+\epsilon}$ for some $\theta_{k,s}<1/2$. As an immediate consequence, we obtain a lower bound for the $L^1$-mean of the M\"obius-twisted exponential sum over short intervals of length at least $N^{0.49685}$. Moreover, we show that further improvements on all of these results would follow immediately from improvements to an $\ell^2$-estimate involving the M\"obius function.

math.NT↗

arXiv2025

On the Moments of Exponential Sums over r-Free Polynomials

Let $\mathbb{F}_q[t]$ denote the ring of polynomials over the finite field $\mathbb{F}_q$. Building off of techniques of Balog and Ruzsa and of Keil in the integer setting, we determine the precise order of magnitude of $k$th moments of exponential sums over $r$-free polynomials in $\mathbb{F}_q[t]$ for all $k>0$. In the supercritical case $k>1+1/r$, we acquire an asymptotic formula using a function field analogue of the Hardy-Littlewood circle method.

math.NT↗

arXiv2024

LLMs are One-Shot URL Classifiers and Explainers

Malicious URL classification represents a crucial aspect of cyber security. Although existing work comprises numerous machine learning and deep learning-based URL classification models, most suffer from generalisation and domain-adaptation issues arising from the lack of representative training datasets. Furthermore, these models fail to provide explanations for a given URL classification in natural human language. In this work, we investigate and demonstrate the use of Large Language Models (LLMs) to address this issue. Specifically, we propose an LLM-based one-shot learning framework that uses Chain-of-Thought (CoT) reasoning to predict whether a given URL is benign or phishing. We evaluate our framework using three URL datasets and five state-of-the-art LLMs and show that one-shot LLM prompting indeed provides performances close to supervised models, with GPT 4-Turbo being the best model, followed by Claude 3 Opus. We conduct a quantitative analysis of the LLM explanations and show that most of the explanations provided by LLMs align with the post-hoc explanations of the supervised classifiers, and the explanations have high readability, coherency, and informativeness.

cs.AI↗

arXiv2023

Algorithms for Chip-Firing on Weighted Graphs

We extend the notion of chip-firing to weighted graphs, and generalize the Greedy Algorithm and Dhar's Burning Algorithm to weighted graphs. For a vertex $q \in V(\Gamma)$, we give an upper bound for the number of linearly equivalent $q$-reduced divisors. Finally, we illustrate a method of finding all maximal unwinnable divisors on weighted graphs.

math.CO↗

arXiv2017

Trustware: A Device-based Protocol for Verifying Client Legitimacy

Online services commonly attempt to verify the legitimacy of users with CAPTCHAs. However, CAPTCHAs are annoying for users, often difficult for users to solve, and can be defeated using cheap labor or, increasingly, with improved algorithms. We propose a new protocol for clients to prove their legitimacy, allowing the client's devices to vouch for the client. The client's devices, and those in close proximity, provide a one-time passcode that is verified by the device manufacturer. This verification proves that the client has physical access to expensive and trusted devices, vouching for the client's legitimacy.

cs.CR↗