Tightness of and counterexamples to several quantum estimates

What the math gives to ML

The paper supplies dimension-free Bohnenblust–Hille bounds converting the supremum of a degree-d polynomial on the complex torus into an ℓp norm of its coefficients, with the sharp exponent p=2d/(d+1). This suggests a concrete regularizer for polynomial or tensorized neural layers: control the function uniformly on normalized phase inputs while explicitly penalizing the coefficient quasi-norm that the inequality identifies as the correct dimension-independent measure. The most credible transfer is to degree-2 or degree-3 polynomial layers, where the coefficient penalty is cheap enough to compare against ordinary Frobenius or ℓ1 regularization and test whether it improves parameter efficiency and stability as input dimension grows.