Comparison of Greedy Angle Decompositions: Harmonic, Dyadic, and General Sequences

For \(\theta = \pi\): at step 0, \(r_0 = \pi \geq \pi/2\), so \(\delta_0 = 1\) and \(r_1 = \pi/2\). At step 1, \(r_1 = \pi/2 \geq \pi/4\), so \(\delta_1 = 1\) and \(r_2 = \pi/4\). By induction, \(\delta_n = 1\) and \(r_{n+1} = \pi/2^{n+1}\) for all \(n\).

The smallest \(\theta\) for which \(\delta_n^{(\mathrm{D})}(\theta) = 1\) is achieved when all earlier digits \(\delta_0 = \cdots = \delta_{n-1} = 0\) and the residual at step \(n\) is exactly \(\alpha_n^{(\mathrm{D})}\). This gives \(\theta_n^{*(\mathrm{D})} = \alpha_n^{(\mathrm{D})} = \pi/2^{n+1}\).

To verify: if \(\theta = \pi/2^{n+1}\), then for all \(k < n\), \(\alpha_k^{(\mathrm{D})} = \pi/2^{k+1} > \pi/2^{n+1} = \theta \geq r_k\), so \(\delta_k = 0\) and the residual remains \(\theta\). At step \(n\), \(r_n = \theta = \pi/2^{n+1} = \alpha_n^{(\mathrm{D})}\), so \(\delta_n = 1\).

However, if we instead centre by the constant \(1/2\), the digits become exactly uncorrelated in the \(L^2([0,\pi])\) inner product, since the binary digits of a uniform random variable in \([0,1]\) are independent. We record this observation for comparison in Section 5.

(a) The residual bound \(r_N \leq \alpha_{N-1}\) follows from the greedy rule exactly as in [Geere, 2026a, Lemma 3.1]: if \(\delta_N = 0\), then \(r_N = r_{N-1} < \alpha_N \leq \alpha_{N-1}\); if \(\delta_N = 1\), then \(r_{N+1} = r_N - \alpha_N < \alpha_{N-1} - \alpha_N \leq \alpha_{N-1}\). Since \(\alpha_n \to 0\) (as a decreasing sequence whose terms must tend to zero for the sum to diverge, or even for a convergent sum), \(r_N \to 0\).

(b)–(c) The proof of [Geere, 2026a, Lemma 4.1] uses only the decreasing property of \((\alpha_n)\) and the greedy rule; it applies verbatim to any decreasing sequence.

(d) At the threshold \(\theta = \theta_n^*\), we have \(\theta_n^* = \alpha_n + \theta_n(\theta_n^*)\) with \(\theta_n(\theta_n^*) \geq 0\).

(e) The quadratic form \(\sum_{n,m} a_n\bar{a}_m K_\alpha(n,m) = \int_0^\pi |\sum_n a_n[\delta_n(\theta) - f(\theta)]|^2\,d\theta \geq 0\).

Harmonic: \(\theta_N \approx \pi \sum_{n : \delta_n = 1} 1/(n+2)\). Since the selected terms contribute \(\theta/\pi\) of the harmonic sum asymptotically (cf. [Geere, 2026a, Lemma 3.3]), \(D_{\mathrm{H}}(N,\theta) \sim (\theta/\pi)\ln N\).

Dyadic: Given the binary expansion interpretation (Theorem 2.2), the density of 1-digits in the binary expansion of \(\theta/\pi\) depends on the specific value of \(\theta\). However, by the equidistribution of binary digits for Lebesgue-typical \(\theta\), the density is approximately \(1/2\) for typical targets. More precisely, \(D_{\mathrm{D}}(N,\theta)/(N+1)\) converges to the asymptotic density of 1-digits in the binary expansion of \(\theta/\pi\), which exists and equals \(1/2\) for Lebesgue-a.e. \(\theta\) (by the normal number theorem).

Power-law: \(\theta_N \approx \pi \sum_{n : \delta_n = 1}(n+2)^{-p}\). Since the full partial sum grows as \(\pi N^{1-p}/(1-p)\) and the selected terms accumulate to \(\theta\), the selection count satisfies \(D_p(N,\theta) \cdot \bar{\alpha}(N) \sim \theta\) where \(\bar{\alpha}(N) \sim \pi N^{-p}\) is the average term size near index \(N\). This gives \(D_p(N,\theta) \sim \theta N^p/\pi\), so \(d_p(N,\theta) \sim \theta N^{p-1}/\pi \to 0\).

(a) Proved in Lemma 2.3. The key is that the dyadic terms decrease geometrically: \(\alpha_k^{(\mathrm{D})} = 2\alpha_{k+1}^{(\mathrm{D})}\), so at the threshold \(\theta = \pi/2^{n+1}\), all earlier terms exceed \(\theta\) and are not selected. Thus \(\theta_n(\theta_n^{*(\mathrm{D})}) = 0\).

(b) Established in [Geere, 2026a, Sections 4–5]. The non-monotonicity arises because the harmonic terms decrease polynomially: \(\alpha_k^{(\mathrm{H})}/\alpha_{k+1}^{(\mathrm{H})} = (k+3)/(k+2) \to 1\), so at the threshold of a later index, some earlier indices may already be selected, adding to the accumulated angle and pushing the threshold higher than \(\alpha_n^{(\mathrm{H})}\) alone.

(c) For the power-law sequence, the ratio \(\alpha_k^{(p)}/\alpha_{k+1}^{(p)} = ((k+3)/(k+2))^p \to 1\) as \(k \to \infty\), similarly to the harmonic case. The number of earlier indices selected at the threshold is of order \(D_p(n, \theta_n^{*(p)})\), which (by Lemma 3.4) grows sub-linearly. The accumulated angle from these selections is bounded by \(\theta_n^{*(p)}\), creating a self-consistent equation. For \(p < 1\), the terms \(\alpha_k^{(p)}\) decrease more slowly, so fewer selections are needed to accumulate a given angle, and the interaction effect is weaker.

(a) Immediate from Lemma 2.3.

(b) For the harmonic sequence, at the threshold \(\theta_n^{*(\mathrm{H})}\), the accumulated angle from earlier selections is \(\theta_n(\theta_n^{*(\mathrm{H})}) = \sum_{k < n,\, \theta_k^{*(\mathrm{H})} \leq \theta_n^{*(\mathrm{H})}} \alpha_k^{(\mathrm{H})}\). When \(\theta_n^{*(\mathrm{H})}\) is not too small, many earlier indices with small angles (large \(k\)) satisfy \(\theta_k^{*(\mathrm{H})} \leq \theta_n^{*(\mathrm{H})}\) and are therefore selected, contributing a sum that can be much larger than \(\alpha_n^{(\mathrm{H})}\). For instance, the threshold \(\theta_0^{*(\mathrm{H})} = \pi/2\) has \(\varepsilon_0^{(\mathrm{H})} = 0\), but for any index \(n\) whose threshold happens to be near \(\pi/2\), many earlier small-angle terms are selected, producing a large excess.

(c) For the power-law sequence with \(p < 1\), the terms decrease more slowly, so at the threshold of index \(n\), fewer earlier indices have angles smaller than \(\theta_n^{*(p)}\) (since \(\alpha_k^{(p)} = \pi/(k+2)^p\) is larger for a given \(k\) compared to the harmonic case). The accumulated angle from the selected earlier indices grows more slowly relative to \(\alpha_n^{(p)}\), bounding the excess ratio.

For the diagonal, substituting \(\theta_n^* = \pi/2^{n+1}\):

• Dyadic: \(\pi/2^N = \varepsilon \implies N = \log_2(\pi/\varepsilon)\).
• Harmonic: \(\pi/(N+1) = \varepsilon \implies N = \pi/\varepsilon - 1\).
• Power-law: \(\pi/(N+1)^p = \varepsilon \implies N = (\pi/\varepsilon)^{1/p} - 1\).

(a) Immediate from Proposition 2.7.

(b) The linearly-centred kernel can be written as \(K_{\mathrm{D}} = \widetilde{K}_{\mathrm{D}} + E\) where \(E\) is a correction matrix arising from the change of centring from constant \(1/2\) to linear \(\theta/\pi\). The correction involves the function \(1/2 - \theta/\pi\), whose \(L^2\) norm is bounded. By Weyl's inequality, the eigenvalues of \(K_{\mathrm{D}}\) differ from those of \(\widetilde{K}_{\mathrm{D}}\) by at most \(\|E\|_{\mathrm{op}}\), where the operator norm of \(E\) is bounded by the \(L^2\) norm of the centring difference times \(\sqrt{N+1}\). For large \(N\), the bulk eigenvalues approach \(\pi/4 + O(1)\).

(a) Strict positive definiteness follows from the linear independence of the step functions \(\delta_n(\theta) - \theta/\pi\) in \(L^2([0,\pi])\). These are linearly independent because the step functions \(\mathbf{1}[\theta \geq \theta_n^*]\) have distinct jump points (the thresholds are all distinct, since each \(\theta_n^*\) depends on all previous selections in a way that generically produces distinct values).

(b) The trace is \(\sum_{n=0}^{N} K_{\mathrm{H}}(n,n) = \Theta(N)\) (since each diagonal entry is \(\Theta(1)\)), so the average eigenvalue is \(\Theta(1)\). The largest eigenvalue is bounded above by the trace, giving \(O(N)\). Due to the oscillatory structure of the thresholds, the off-diagonal mass is distributed, preventing any eigenvalue from growing faster than \(O(N)\).

(c) The non-concentration follows from the non-trivial correlation between different indices: unlike the dyadic case, \(K_{\mathrm{H}}(n,m)\) is not zero or negligible for many pairs \((n,m)\), so the kernel matrix is not close to a scalar times the identity.