A centuries-old riddle about the denominators of fractions just fell apart thanks to a new theory about high-dimensional matrices.
Zaremba's conjecture has frustrated mathematicians for generations with its claim about which numbers can appear in the bottom of continued fractions. This new proof uses expansion theory within special linear groups to confirm the conjecture is actually true. The breakthrough relies on understanding how certain types of grids grow and interact in modular arithmetic. Solving this removes a massive roadblock in number theory and changes how we understand the fundamental distribution of rational numbers. It provides a definitive answer to a question that was previously thought to be beyond the reach of current methods.
Expansion in SL2(Z/qZ) and Zaremba's conjecture
arXiv · 2605.02518
We establish an expansion theory for $\text{SL}_2(\mathbb Z/q\mathbb Z)$. Incorporating this into a framework recently developed by Shkredov, we confirm Zaremba's conjecture.