arXiv subjects

David Burt

Publications and source records attributed to David Burt.

5 recordsLinked to original sources

arXiv2026

Optomechanical Levitation and Control of High Aspect Ratio Silicon Nanorods

Nano- and micro-particles levitated by optical, electrical or magnetic fields are a new frontier in precision sensing and for tests of fundamental physics. For optically levitated anisotropic particles it is possible to control their translation, alignment and rotation. We report on the levitation and characterization of nanofabricated, high uniformity, high aspect ratio, high refractive index silicon cylinders, with diameters as low as 50 nm and lengths up to 1500 nm. We are able to tune their oscillation frequencies from 10 kHz to over 1 MHz, and exert huge optical torque to generate high rotation rates. These optically levitated silicon nanorods will enable precision torque sensing, and when pushed to smaller sizes, tests of quantum physics through the generation of angular momentum superposition states.

physics.optics↗

arXiv2026

Field validation of GNSS-independent positioning enhancement using a wearable ultra-stable quantum magnetometer

Increasing the resilience of positioning systems that currently rely on Global Navigation Satellite System (GNSS) signals can be achieved by incorporating stable and sensitive measurements of the permanent crustal anomalies in the Earth's magnetic field. We have realised this concept using an in-house-developed, wearable, Free-Induction-Decay Optically Pumped Magnetometer (FID-OPM) to carry out precise and stable measurements of the geomagnetic field in a walking trial. We present an end-to-end validation, including qualification of FID-OPM performance, alongside quantification of improvement in accuracy when data from this sensor is added to a dead-reckoning estimation of position. Using our wearable sensor system we achieve a Beckmann-distributed radial positioning error of 2.24 m over a route exceeding 500 m in length and spanning approximately 360 s.

physics.atom-ph↗

arXiv2015

Crescent configurations

In 1989, Erd\H{o}s conjectured that for a sufficiently large $n$ it is impossible to place $n$ points in general position in a plane such that for every $1\le i \le n-1$ there is a distance that occurs exactly $i$ times. For small $n$ this is possible and in his paper he provided constructions for $n\leq 8$. The one for $n=5$ was due to Pomerance while Pal\'{a}sti came up with the constructions for $n=7,8$. Constructions for $n=9$ and above remain undiscovered, and little headway has been made toward a proof that for sufficiently large $n$ no configuration exists. In this paper we consider a natural generalization to higher dimensions and provide a construction which shows that for any given $n$ there exists a sufficiently large dimension $d$ such that there is a configuration in $d$-dimensional space meeting Erd\H{o}s' criteria.

math.CO↗

arXiv2013

Benford's Law and Continuous Dependent Random Variables

Many mathematical, man-made and natural systems exhibit a leading-digit bias, where a first digit (base 10) of 1 occurs not 11\% of the time, as one would expect if all digits were equally likely, but rather 30\%. This phenomenon is known as Benford's Law. Analyzing which datasets adhere to Benford's Law and how quickly Benford behavior sets in are the two most important problems in the field. Most previous work studied systems of independent random variables, and relied on the independence in their analyses. Inspired by natural processes such as particle decay, we study the dependent random variables that emerge from models of decomposition of conserved quantities. We prove that in many instances the distribution of lengths of the resulting pieces converges to Benford behavior as the number of divisions grow, and give several conjectures for other fragmentation processes. The main difficulty is that the resulting random variables are dependent. We handle this by using tools from Fourier analysis and irrationality exponents to obtain quantified convergence rates as well as introducing and developing techniques to measure and control the dependencies. The construction of these tools is one of the major motivations of this work, as our approach can be applied to many other dependent systems. As an example, we show that the $n!$ entries in the determinant expansions of $n\times n$ matrices with entries independently drawn from nice random variables converges to Benford's Law.

math.PR↗

arXiv2007

Irrationality measure and lower bounds for pi(x)

In this note we show how the irrationality measure of $\zeta(s) = \pi^2/6$ can be used to obtain explicit lower bounds for $\pi(x)$. We analyze the key ingredients of the proof of the finiteness of the irrationality measure, and show how to obtain good lower bounds for $\pi(x)$ from these arguments as well. While versions of some of the results here have been done by other authors, our arguments are more elementary and yield a lower bound of order $x/\log x$ as a natural boundary.

math.NT↗