= Uniform equidistribution of an irrational skew shift
{title2=$A_Ng\to\int g\,dm_2$ uniformly}
For an <irrational skew shift>, the averages of every continuous function $g$ converge uniformly in the starting point to $\int g\,dm_2$, with $m_2$ normalized <Lebesgue measure>. Nonconstant <Fourier basis> characters have either linear or quadratic phases. Linear phases are bounded <geometric series>; for quadratic phases the <Van der Corput inequality for finite scalar sequences> reduces to linear correlations of irrational frequency, uniformly in the starting point. Approximation by <trigonometric polynomials> proves the assertion and identifies every invariant <Borel probability measure> as $m_2$.
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