Idempotent concentration audit

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The specific invalid inference

After reading the supplied digest, the relevant passages of the primary 2025 preprint were checked: Lemma 5, its preceding definition on page 6, and equation (27) with its concluding sentence on page 9. Equation (27) gives full concentration on neighborhoods of the identity, then invokes the bound $c_{2k}\le1/2$ as a contradiction.

For idempotents (polynomials with coefficients zero or one), write

$$R_p(Q,E)=\frac{\int_E|Q|^p}{\int_{\mathbb T}|Q|^p}.$$

The relevant global constant is an infimum over nonempty symmetric open sets $E$ of $\sup_Q R_p(Q,E)$. A bound on that infimum is not a bound for every individual $E$.

The distinction is explicit in the primary Bonami–Révész paper: the sentence after Proposition 9 on page 5 states that Dirichlet kernels give full concentration at zero for $p>1$. The even-exponent obstruction discussed on page 10 comes from concentration at $1/2$ in $\mathbb R/\mathbb Z$, not at zero. These passages and their hypotheses were checked; this is not acceptance of the claimed resolution.

The concluding concentration requires only bounded normalized norms

Proved lemma. Let $L_N$ be a Littlewood polynomial with $N$ coefficients, let $D_N(z)=\sum_{j<N}z^j$, and let $Q_N=(D_N+L_N)/2$, an idempotent. For fixed $p>2$, suppose $\|L_N\|_p\le K\sqrt N$. Then

$$\left\| \frac{|Q_N|^p}{\|Q_N\|_p^p} -\frac{|D_N|^p}{\|D_N\|_p^p} \right\|_1 =O_{p,K}(N^{1/p-1/2}). \tag{1}$$

In particular $R_p(Q_N,U)\to1$ for every fixed neighborhood $U$ of zero, without any near-unit upper-flatness constant.

To prove this, set $B_N=2Q_N$, $a_N=\|D_N\|_p$, and $b_N=\|B_N\|_p$. The elementary Dirichlet-kernel bounds give $a_N\asymp_p N^{1-1/p}$: use $|D_N(e^{it})|\le\min(N,C/|t|)$ for the upper estimate, and $|D_N(e^{it})|\ge cN$ on $|t|\le c/N$ for the lower estimate. Since $\|B_N-D_N\|_p\le K\sqrt N=o(a_N)$, the reverse triangle inequality gives $b_N/a_N\to1$, and

$$\left\|\frac{B_N}{b_N}-\frac{D_N}{a_N}\right\|_p \le\frac{2\|L_N\|_p}{b_N} =O_{p,K}(N^{1/p-1/2}).$$

For unit $L^p$-norm functions $f,g$, the pointwise power inequality and Hölder give $\||f|^p-|g|^p\|_1\le2p\|f-g\|_p$. This proves (1). The normalized Dirichlet mass outside $U$ tends to zero because $D_N$ is uniformly bounded there while $a_N^p\asymp N^{p-1}\to\infty$.

For a concrete valid family, the classical Rudin–Shapiro recursion supplies $\|L_N\|_\infty\le\sqrt{2N}$ at dyadic lengths. Its associated idempotents therefore have precisely this full concentration at zero. Thus that concentration cannot be the claimed contradiction.

This audit rejects the stated inference. It does not disprove the claimed theorem itself.