Diophantine Approximation and Banach Space Geometry

We propose a new research program connecting Diophantine approximation in transcendental number theory with geometric and analytic properties of Banach spaces. The key idea is to study embeddings of real numbers with prescribed irrationality exponent μ(x) into Banach spaces via characteristic functions, and to examine operator norm profiles that depend continuously on real parameters. Three main contributions are presented. First, we establish an explicit formula for the Hausdorff dimension of sets of characteristic functions χ[0,x] with x of given irrationality exponent when embedded into Lp([0,1]), showing that the dimension is scaled by the snowflake exponent 1/p. Second, we prove a Diophantine Stability Theorem for operator norms depending on real parameters, showing that the irrationality exponent of the parameter controls that of the operator norm value. Finally, we derive a quantitative Banach–Diophantine Correspondence Theorem connecting Hölder regularity of analytic families with amplification of irrationality exponents, and illustrate it through the Banach–Mazur distance between Lp spaces.