= Cut norm and rectangle fourth-moment equivalence
{title2=$\|f\|_{\mathrm{cut}}\leq\|f\|_\square\leq(4\|f\|_{\mathrm{cut}})^{1/4}$}
For a real function $f$ bounded by one on a finite Cartesian product, the fourth power of its <box norm> is the mean of $f(x,y)f(x,y')f(x',y)f(x',y')$. Two applications of the <Cauchy-Schwarz inequality> bound every rectangular discrepancy by the <box norm>. Conversely, decompose each bounded real test function into positive and negative parts and use the <layer cake representation>. Its bilinear pairing with $f$ is at most four times the <cut norm>. Applying this to the row and column tests furnished by fixed $x',y'$ bounds the fourth moment by four times the <cut norm>. These dimension-independent estimates make the two notions of small discrepancy equivalent.
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