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In this paper, we investigate the fractal uncertainty principle (FUP) fordiscrete Cantor sets, which are determined by an alphabet from a base ofdigits. Consider the base of M digits and the alphabets of cardinality A suchthat all the corresponding Cantor sets have a fixed dimension 0<log A/logM<2/3. We prove that the FUP with an improved exponent over Dyatlov-Jin(arXiv:1608.02238) holds for almost all alphabets, asymptotically as M tends toinfinity. Our result provides the best possible exponent when the Cantor setsenjoy either the strongest Fourier decay assumption or strongest additiveenergy assumption. The proof is based on a concentration of measure phenomenonin the space of alphabets.

Fractal uncertainty principle for discrete Cantor sets with random alphabets