Kolmogorov polynomial blocks — recorded construction lemma

For every integer $M\ge 1$ there exist a nonnegative trigonometric polynomial $g_M$ on $\mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$ and a measurable set $A_M\subseteq \mathbb{T}$ such that, for normalized Haar measure,

$\mathrm{\parallel }{g}_M{\mathrm{\parallel }}_1=1,m(A_M)>1-2^{-M},{\inf }_{x\in A_M}{\sup }_{N\ge 1}\mathrm{\mid }{S}_N{g}_M(x)\mathrm{\mid }>2^M\mathrm{.}$

The supremum is of the same partial sums as Carleson maximal partial-sum operator. This is the integer version of Grafakos, Lemma 4.2.4, including nonnegativity from its construction. The proof is not supplied here. “Block” does not mean disjoint frequency support.

Grafakos's Lemma 4.2.2 aligns phases: if $1,x_1,\dots ,x_n$ are linearly independent over $\mathbb{Q}$, then for any unimodular $z_j$ and $\epsilon >0$ some integer $L$ satisfies $\mathrm{\mid }{e}^{2\pi iL{x}_j}-z_j\mathrm{\mid }<\epsilon$ for all $j$. Its Fourier-averaging argument is not established locally.

Lemma 4.2.3 constructs atomic probability measures ${\mu }_n$ with ${\sup }_{L\ge 1}\mathrm{\mid }{D}_L\ast {\mu }_n\mathrm{\mid }\ge c\log n$ almost everywhere, for an absolute $c>0$. Rational independence supplies simultaneous alignment of the kernel terms. Lemma 4.2.4 then selects a finite maximal truncation and smooths ${\mu }_n$ with a Fejer kernel. Positivity and mass one survive this smoothing, while the finitely many partial sums remain close enough to preserve the required height. These are descriptions of unproved source machinery, not local facts available as dependencies.