Document Type

Presentation Abstract

Presentation Date

11-24-2025

Abstract

How can we visualize the convergence of rational approximations to any real number? We will use continued fractions, the iterative process that generates rational approximations, and the Stern-Brocot Diagram, which defines the unique structure and relationship between rationals in lowest terms. A Farey Recursive Function (F) is defined on the points of the Stern-Brocot Diagram, where the F-values of points lying on any line in the diagram satisfy a 2-term linear recurrence relation. For example, the Fibonacci sequence on the nonnegative integers can be extended across all rational numbers by a Farey Recursive Function.

But what happens when we look at the F-values of points that lie on a Euclidean line that is not a line in the diagram?

This presentation introduces Cross Recursion, proving that sequences of F-values along these Euclidean lines not in the diagram still follow linear recurrence relations. Our main result shows that the F-values of points lying on a Euclidean line passing through (p/q,1/q) and (a/b,0) is an interleaving of φ(w) linear recursive sequences, where φ(w) is the Euler totient function and w=|pb − qa|. Each of these φ(w) sequences has an order at most w+1. Furthermore, the F-values of points lying on a horizontal line y=1/w is an interleaving of φ(w) linear recursive sequences, where each of these φ(w) sequences is a bi-infinite sequence with an order at most w+1.

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Additional Details

November 24, 2025 at 3:00 p.m. Math 103

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