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On a distinctive property of Fourier bases associated with $N$- Bernoulli Convolutions

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For N-Bernoulli convolutions, a rescaled Fourier spectrum fails to be an orthonormal basis precisely when a digit-restricted fractal hits an integer.

desk verdict A solid, genuinely new treatment of complete scaling numbers for N-Bernoulli convolutions; the main theorems hold together, and the paper deserves refereeing despite some rough presentation. read the letter →

arxiv 2506.00364 v1 pith:FFYY2JJV submitted 2025-05-31 math.CA

Zi-Chao Chi , Xing-Gang He , Zhi-Yi Wu This is my paper · ORCID
classification math.CA MSC 28A8042C0511A0742A65
keywords N-BernoulliconvolutioncompletenumberspectralpairFourierorthonormalbasisself-similarmeasuredensityofscalingsorderaninteger
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

An N-Bernoulli convolution is a fractal measure built by repeatedly spreading equal mass over N equally spaced digits and rescaling. For measures of this type that are spectral, meaning some countable family of exponentials forms an orthonormal basis, this paper asks which scalings t of that spectrum preserve the basis property. It proves a short, checkable criterion: for integers t coprime to N, the scaled basis fails exactly when the digit-restricted fractal attractor T(b,C) also meets the integers after scaling by t. This number-theoretic reformulation lets the authors prove the general conjecture that infinitely many such complete scalings exist (and infinitely many incomplete ones), give upper and lower density bounds for complete scalings, and settle a prime-power question in the two-digit case.

What carries the argument

The load-bearing object is the intersection T(b,tC)∩Z. The set T(b,tC) is the fractal attractor of the maps x↦(x+t c)/b with c in the digit set C=(-N/2,N/2]∩Z, and the condition that this intersection contains a non-zero integer is equivalent, through the recursion x_{k+1}=(x_k+t c_k)/b, to the existence of a non-zero integer periodic orbit. Theorem 2.5 identifies this condition with the failure of tΛ(b,C) to be a spectrum, and the rest of the paper reads consequences out of it: the order O_b(t) of b modulo t controls the minimal periods of such orbits (Lemma 3.3, Theorem 4.1), and the Erdős–Murty lower bound on orders of primes modulo b drives the density results. The cyclic group generated by b modulo t also appears in Theorem 4.4, giving sufficient conditions for completeness when certain small residues occur in that group.

What would settle it

Take a large prime t coprime to N for which T(b,tC)∩Z={0} (an exact finite check, since the attractor is a nested union of intervals) and evaluate the Jorgensen–Pedersen sum Q_{tΛ(b,C)}(ξ) numerically at many points ξ; if any value is strictly below 1, the pair is not spectral and the criterion fails. Conversely, for a t with T(b,tC)∩Z non-trivial, explicitly running the recursion x_{k+1}=(x_k+t c_k)/b and verifying that all c_k lie in C and all x_k are integers confirms the incomplete side; a single mismatch would void the example, and one t with empty intersection that is still incomplete would refute Theorem 2.5.

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Extended reading notes

Core claim

The central claim is Theorem 2.5: for the spectral pair (µ_{b,qD}, Λ(b,C)) with b=qN and q≥2, an integer t with gcd(t,N)=1 is an incomplete number precisely when T(b,tC)∩Z ≠ {0}. Here T(b,A) is the self-similar compact set of all sums Σ $b^{{-k}}$ a_k with digits a_k∈A. The theorem also recasts this failure of spectrality as the existence of a non-zero integer periodic sequence satisfying x_{k+1}=(x_k+t c_k)/b with c_k∈C. Building on this equivalence, the paper proves that infinitely many primitive complete and incomplete numbers exist (Theorem 1.7), that the set of complete numbers has upper density at most c·φ(N)/N for a constant 0<c<1 depending on b while having positive lower density among primes when q>N (Theorem 1.4), that prime powers p^n are complete in the case N=2, b=2^r whenever rO_2(p) is even or r∤(p-1) (Theorem 1.8), and that sufficiently large complete prime powers imply completeness of all smaller powers of those primes (Theorem 1.9).

Load-bearing premise

The whole characterization leans on an imported theorem asserting that a scaled spectrum fails to be a basis exactly when a certain integer recursion has a non-zero repeating orbit, and the paper treats that external theorem as a black box without reproving it.

Editorial extensions

If this is right

  • The completeness of an integer t for this family of spectral pairs becomes a finite, checkable condition: one only needs to determine whether T(b,tC) contains a non-zero integer.
  • Theorem 1.7 establishes the general conjecture in this setting: there are infinitely many primitive complete and primitive incomplete numbers, not just finitely many sporadic examples.
  • Theorem 1.4 shows that the complete numbers form a set of upper density strictly less than 1, controlled by c·φ(N)/N, while whenever q>N they still have positive lower density on the prime scale.
  • Theorem 1.8 extends the known two-digit result to b=2^r for all r≥2, giving a clean number-theoretic condition under which every power p^n of an odd prime p≠2^r-1 remains a complete number.
  • Theorem 1.9 shows that completeness propagates downward in prime-power exponents once a sufficiently large mixed power is complete.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the criterion is an intersection of a self-similar set with Z, the set of complete numbers for fixed b,N is likely recognized by a finite automaton on a suitable digit representation of t; a computational census of small t would test this directly.
  • The same criterion may extend to other self-similar spectral measures with integer digit sets beyond consecutive digits, in which case the order-based machinery and density bounds would transfer to those families.
  • The upper density bound suggests a heuristic that the typical complete number has density exactly c·φ(N)/N; identifying the constant c explicitly would sharpen the probabilistic picture of which scalings preserve bases.
  • The complete and incomplete numbers give concrete families of scalings for which mock Fourier series diverge or converge at 0, connecting the number-theoretic classification to the convergence results for such series.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper studies the 'complete number' problem for N-Bernoulli convolutions: for the spectral pair (μ_{b,qD}, Λ(b,C)) with b=qN, D={0,...,N-1}, and C=(-N/2,N/2]∩Z, it asks which t preserve spectrality when Λ is replaced by tΛ. The central result is Theorem 2.5, an equivalence between incompleteness of t and the existence of a non-zero integer periodic sequence for (b,tC), equivalently the condition T(b,tC)∩Z≠{0}. On this basis the paper derives density bounds for complete numbers (Theorem 1.4), proves that there are infinitely many primitive complete and incomplete numbers (Theorem 1.7), gives sufficient conditions for composite numbers to be complete (Theorem 5.7), and analyzes the case N=2, b=2^r (Theorems 1.8 and 6.10). The proofs rely on external results of Łaba-Wang, Jorgensen-Pedersen, Strichartz, and Erdős-Murty; I checked the key scaling in Theorem 2.5 and the Łaba-Wang hypotheses appear to be met when gcd(t,N)=1.

Significance. If the results are correct, they constitute a substantial extension of the earlier work of Dutkay-Haussermann and Dutkay-Kraus on Cantor measures, replacing ad-hoc case analysis with a number-theoretic criterion and producing new density and infinitude statements. The paper also provides concrete, computer-checkable examples of incomplete primes, which gives the main criterion falsifiable content. The central derivation is not circular: it imports standard theorems rather than fitting parameters to data. The contribution is specialized to fractal harmonic analysis, but within that area it addresses an open conjecture and a natural question for a non-trivial family of measures. The main weaknesses are presentation: some proofs are not readable as typeset, one theorem uses an undefined symbol, and two statements of the same result disagree.

major comments (3)
  1. [Section 3, Proposition 3.6] The proof of Proposition 3.6 is not verifiable as printed. The displayed inequalities such as 'logN 1/2 p 1/2+ϵ(p)−log q 2p 1+ϵ(p) 2+ϵ(p)' do not specify bases, parentheses, or the intended relation between the two logarithms, and the step 'there exists k_0 such that ...' does not follow transparently from the preceding estimate. Since this proposition supplies the entire proof of the lower-density assertion in Theorem 1.4, the argument must be rewritten with explicit logarithmic inequalities and a carefully justified choice of k_0; the later line containing 'p/q^{k−0}' should also be corrected to a proper subscript.
  2. [Section 4, Theorem 4.4] In the proof of Theorem 4.4 the symbol m in the inequality '|a|≤q−1<(b−1)/m' is never defined, so the proof of part (i) is incomplete as written. The context strongly suggests that m should be N, and the following paragraph should read 'q=2' rather than 'b=2N', but this must be stated explicitly and the estimates must be checked for both parities of N. Because Theorem 1.7 uses Theorem 4.4 to construct infinitely many primitive complete numbers, this is a load-bearing gap.
  3. [Section 6, Theorem 1.8 and Theorem 6.10] The introduction states Theorem 1.8 with the hypothesis 'r∤(p−1)', while Theorem 6.10 at the end of Section 6 is stated with 'r∤O_2(p)'. These are not the same condition, and the manuscript does not explain that the former follows from the latter via O_2(p)|p−1. In addition, the paragraph before Lemma 6.8 says the case rO_2(p) odd is 'guess[ed]' and 'cannot prove it', but Lemma 6.8 immediately proves the subcase r∤O_2(p). The genuinely open case (rO_2(p) odd and r|O_2(p)) should be stated explicitly, and the two theorem statements should be reconciled.
minor comments (6)
  1. [Section 2, Theorem 2.5 and Corollary 2.9] The denominators in Theorem 2.5(iii), Corollary 2.9, and Lemma 3.7 are printed as 'bn−1' and 'bk−1' without carets; they should read b^n−1 and b^k−1. Since Theorem 2.5 is the main criterion, these formulas should be typeset unambiguously.
  2. [Section 3, Lemma 3.2] In the proof of Lemma 3.2, 't(m−1)/(b^k(b−1))' should be 't(N−1)/(b^k(b−1))', and the interval notation should be cleaned up.
  3. [Section 4, Corollary 4.5] In Corollary 4.5 and its proof, 'dmb^k±1' should be 'dN b^k±1'.
  4. [Section 5, Theorem 5.7] In the proof of Theorem 5.7, 'since we suppose m≥3' should be 'since we suppose N≥3'; the displayed inequality does not depend on the number of primes m.
  5. [Introduction and Section 6] The notation O_2(p) is used in Theorem 1.8 before its definition in Section 3; please define it earlier or add an explicit forward reference to Definition 3.1.
  6. [Section 3, definition of Γ*] In the display after the definition of Γ*(x), the limit should be written with #Γ*(x) rather than Γ*(x) alone.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central criterion imports an external Łaba–Wang theorem, and the self-citations are contextual only.

full rationale

The derivation chain is self-contained against external benchmarks. The load-bearing step, Theorem 2.5, imports the non-spectrality characterization from Łaba and Wang [17, Theorem 1.3], an independent external theorem whose hypotheses are met in this setting: b=qN, the candidate spectrum tqΛ(b,C) is contained in Z, and the digit set D has N elements that are distinct modulo b when gcd(t,N)=1. Lemma 2.2 is a direct scaling equivalence proved from the Jorgensen–Pedersen criterion, not a definitional shortcut. No parameter is fitted to data and then renamed as a prediction; the density estimates (Theorem 1.4) and the infinitude results (Theorem 1.7) are derived from Erdős–Murty, order-of-b estimates, and elementary inclusion–exclusion. The self-citations to [14], [15], and [22] appear in contextual references to prior results, not as the sole justification of any central claim, and they do not import a uniqueness theorem that forbids alternatives. The remaining weaknesses are typographical or explicitly open cases, such as the case r and O_2(p) both odd with r|O_2(p), which are correctness concerns rather than circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper relies on established external theorems, not on new postulates or fitted parameters. No free parameters are introduced; the constant c in Theorem 1.4 is a product over primes, not fitted.

assumptions (6)
  • standard math Prime number theorem: π(x) ~ x/log x.
    Used in Section 3 to convert the Erdos-Murty bound into a density statement (Theorem 1.4 and Proposition 3.6).
  • standard math Erdos-Murty theorem: for a>1, for all but o(x/log x) primes p≤x, O_a(p) ≥ p^{1/2+ε(p)}.
    External analytic number theory result used to prove the lower density bound in Theorem 1.4.
  • domain assumption Łaba-Wang Theorem 1.3: for self-similar measures with consecutive digit sets, non-spectrality is equivalent to existence of a non-zero integer periodic sequence.
    The core equivalence in Theorem 2.5 links complete numbers to periodic sequences.
  • domain assumption Jorgensen-Pedersen criterion: (µ,Λ) is spectral iff Q_Λ(ξ)≡1.
    Used throughout to translate spectrality into orthogonality and tiling conditions.
  • domain assumption Strichartz spectrality criterion: if Z(M_{b^{-1}qD}) ∩ T(b,C)=∅, then (µ_{b,qD}, Λ(b,C)) is a spectral pair.
    Establishes the base spectral pair in Lemma 2.3 from [19].
  • domain assumption The mask function zero set formula Z(M_D)=N^{-1}(Z\NZ) and the product formula for the Fourier transform of self-similar measures.
    Used in Section 2 to compute zeros and prove Proposition 2.4 and Theorem 2.5.

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Pith. "Pith review of On a distinctive property of Fourier bases associated with $N$- Bernoulli Convolutions." pith.science (2026). https://pith.science/paper/FFYY2JJV

@misc{pith2026250600364,
  author       = {Pith},
  title        = {Pith review of: On a distinctive property of Fourier bases associated with $N$- Bernoulli Convolutions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FFYY2JJV}},
  note         = {Machine review of arXiv:2506.00364}
}
abstract

A distinctive problem of harmonic analysis on $\R$ with respect to a Borel probability measure $\mu$ is identifying all $t\in\R$ such that both \[\left\{e^{-2\pi i\lambda x}: \lambda\in\Lambda\right\}\quad\text{and}\quad \left\{e^{-2\pi i\lambda x}: \lambda\in t\Lambda\right\}\] form orthonormal bases of the space $L^2(\mu)$. Currently, this phenomenon has been observed only in certain singular measures. It is deeply connected to the convergence of Mock Fourier series with respect to the aforementioned bases. In this paper, we apply classical number theory to solve the general conjecture and basic problems in this field within the setting of $N$-Bernoulli convolutions, which extend almost all known results and give some new ones.

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Reference graph

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