The distribution of minimal fractions on the Stern-Brocot tree
Résumé
The Stern-Brocot (SB) tree enumerates all irreducible rationals, uniquely positioned with ties to continued fractions and Christoffel words. In this work, we study a quotiented SB tree, focusing on fractions at levels multiple of 3 and their presence on minimal paths defined by the second-order balancedness of Christoffel words. We describe the distribution of all fractions belonging to a minimal path at these levels, detailing their positions, and provide a comprehensive study of the gaps between consecutive fractions, revealing structural patterns tied to the quotiented tree. In prior work, fractions Fn F n+1 with Fn from the Fibonacci sequence were found to be on a minimal path (zigzag path). Analogously, we study four other minimal paths that define the boundary of the set of all minimal paths.
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