Fourier Analytic Methods to Count Rational Points near Manifolds

Counting rational points (fractions) near geometric objects has wide ranging applications across mathematics and its interdisciplinary domains. In geometry, it provides insights into the arithmetic properties of surfaces. In number theory, it contributes to understanding the distribution of rational solutions to Diophantine inequalities. The study of rational points near smooth surfaces has seen rapid development in the recent years. To make progress on some of the long-standing questions in this area, a deep understanding of the microlocal Fourier analytic behavior of these manifolds is needed. This specific type of interplay between harmonic analysis and number theory is very recent and not widely understood. The Principal Investigator (PI)'s long term research objective is to apply techniques from harmonic analysis and homogeneous dynamics (geometry of numbers) to make substantial progress on a variety of counting problems. The project provides research training opportunities for graduate students. Consider a smooth bounded manifold; for example, a compact piece of a sphere with non-vanishing Gaussian curvature; or a space curve like the helix which is not contained in any plane. The PI is interested in counting the number of rational points, with denominators of bounded size, in close proximity to such manifolds. This project aims to advance the existing knowledge in several directions: 1. Establish an asymptotic for the number of such rational points in a sharp range of distance from a convex hypersurface or a non-degenerate curve by using a precise understanding of the Fourier transform of their surface measure. Very little is known at the moment; 2. Use a combination of Fourier analytic techniques and methods from the geometry of numbers to count rational points near general smooth manifolds of arbitrary dimension under mild geometric conditions; 3. Establish how far the above methods can go in terms of the range of proximity to the manifold before local algebraic considerations become dominant; and 4. Use the above estimates to answer questions on Multiplicative Diophantine Approximation on analytic manifolds, which are currently wide open. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria. NSF Award ID: 2554830 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Rajula Srivastava | Institution: University of Wisconsin-Madison, MADISON, WI | Award Amount: $200,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2554830 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2554830.html