# Fourier Analytic Methods to Count Rational Points near Manifolds

> **NSF 01002627DB NSF RESEARCH & RELATED ACTIVIT** · University of Wisconsin-Madison (WI) · $200,000

## Abstract

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

## Key facts

- **NSF award ID:** 2554830
- **Awardee organization:** University of Wisconsin-Madison (WI)
- **SAM.gov UEI:** LCLSJAGTNZQ7
- **PI:** Rajula Srivastava
- **Primary program:** 01002627DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $200,000
- **Funds obligated:** $200,000
- **Transaction type:** Standard Grant
- **Period:** 06/15/2026 → 05/31/2029

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2554830

## Citation

> US National Science Foundation, Award 2554830, Fourier Analytic Methods to Count Rational Points near Manifolds. Retrieved via AI Analytics 2026-06-07 from https://api.ai-analytics.org/grant/nsf/2554830. Licensed CC0.

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