Isaac's Condition (TU) is false

Source abstract

In 2006 Isaac reduced the simple normality to base $2$ of $\sqrt s$, for $s$ not a perfect square, to a property of dyadic expansions that he called Condition~(TU): the upper frequency of the digit~$1$ in $\sqrtν$, along any subsequence, depends only on the tail of the expansion of $ν$. We show that Condition~(TU) is false, already for the full sequence. We give two proofs. The first is elementary and turns Isaac's own reduction against the condition. The second uses the theorem of Kaufman and Mosquera--Shmerkin on the Fourier decay of smooth images of self-similar measures, and gives counterexamples for almost every point of a Cantor measure. All results are machine-verified in Lean~4; the second proof uses the two cited theorems as hypotheses.