On structured cosine sums and applications

Qin Xue

REVIEW 2 major objections 3 minor 18 references

On structured cosine sums and applications

T0 review · 2 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For odd N, one nonzero Fourier coefficient can force two small multisets to be identical, and this rigidity bounds eigenvalue multiplicities of cyclic Cayley graphs.

desk verdict New rigidity theorem and a sharp two-cosine bound are real, but the vanishing-sum classification has a self-conjugate proof gap that needs fixing before acceptance. read the letter →

arxiv 2607.20907 v1 pith:ALQIQ7Y7 submitted 2026-07-23 math.NT math.SP

Qin Xue This is my paper · ORCID

classification math.NTmath.SP MSC 11L0311R1805C5011T2220C05

keywords vanishingsumsofrootsunitycosineFourierrigiditygroupringcyclicCayleygraphseigenvaluemultiplicityGaussianperiodstrigonometricDiophantineequations

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

The paper studies finite sums of cosines of rational angles attached to a multiset S in Z/NZ by rewriting them as evaluations $2C_S(k)=ev_N(\operatorname{Sym}(kS))$ in the integral group ring $\mathbb{Z}[\mathbb{Z}/N\mathbb{Z}]$. Its central result is a small-weight Fourier rigidity: for odd N with smallest prime divisor p, two multisets of weight at most p that share a nonzero Fourier coefficient are the same multiset, and the same holds for symmetric multisets of equal weight at most 2p. This rigidity yields concrete spectral consequences for cyclic Cayley graphs: nonzero eigenvalues of small-support symmetric generating sets have multiplicity at most |S|, and the first cosine sum is either 0 or irrational for odd composite N. The paper also gives block-decomposition criteria for vanishing cosine sums and, for square-free N, an exact description of the spectrum as scaled Gaussian periods when S is a subgroup of the unit group.

What carries the argument

The load-bearing object is the evaluation map $ev_N:\mathbb{Z}[\mathbb{Z}/N\mathbb{Z}]\to\mathbb{Z}[\zeta_N]$, together with the identity $2C_S(k)=ev_N(\operatorname{Sym}(kS))$, which converts cosine sums into vanishing-sum questions in the integral group ring. The argument for rigidity reduces N to its square-free radical M, decomposes $\mathbb{Z}/M\mathbb{Z}$ by the Chinese remainder theorem along the smallest prime factor p, and slices the group ring into fibers; the comparison lemma shows that equal evaluations force the fiber evaluations to differ by a constant, after which weights and the known classification of minimal vanishing sums force the fibers to agree. The vanishing criteria use the same classification: for two-prime moduli, minimal vanishing sums are exactly the prime subgroups up to translation, so symmetric vanishing multisets decompose into the three block types $A(u;N/p)$, $A(v;N/q)$, and $D_{p,q}(N)$. For the two-term fiber result, the proof uses the six-form list of conjugation-stable vanishing sums of weight 8 to recover the four-cosine classification and then sharpens it to the bound $|R(c)|\leq 3$ with explicit equality cases.

What would settle it

Enumerate all multisets on $\mathbb{Z}/15\mathbb{Z}$ of weight at most 3 and compare their nonzero evaluations $ev_{15}$; two distinct multisets with the same nonzero value would refute Theorem 4.1. Enumerate symmetric multisets of weight 6 on $\mathbb{Z}/15\mathbb{Z}$: a distinct pair with the same nonzero evaluation would refute the symmetric form.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.1. Let N be odd and let p be its smallest prime divisor. If X and Y are multisets on $\mathbb{Z}/N\mathbb{Z}$ with max weight at most p and $ev_N(X)=ev_N(Y)\neq 0$, then $X=Y$; if X and Y are symmetric with equal weight at most 2p and the same nonzero evaluation, then $X=Y$. Because Galois automorphisms move $ev_N(1)$ to every primitive Fourier coefficient, equality at one nonzero primitive coefficient forces equality of all primitive coefficients, so the statement is genuinely a Fourier rigidity. The nonzero condition and the weight bounds are each necessary, as the paper shows by examples where zero evaluation or weight $2p+1$ breaks rigidity. A companion vanishing theorem says that for $N=p^a q^b$ with distinct odd primes, any symmetric vanishing multiset is a finite sum of the explicit blocks $A(u;N/p)$, $A(v;N/q)$, and $D_{p,q}(N)$; for weight equal to twice the smallest prime divisor, vanishing occurs exactly when the multiset is a single block $A(u;N/p_1)$.

Load-bearing premise

The arguments depend on the completeness of the previously published lists of minimal ways roots of unity can sum to zero, and on one case in the induction (where a vanishing piece equals its own reflection) being closable; if either assumption fails, the paper's vanishing and rigidity conclusions are not established.

Editorial extensions

If this is right

For odd composite N with smallest prime factor p, any symmetric generating set S with $|S|\leq 2p$ that contains a unit has the property that every nonzero eigenvalue $\mu$ satisfies $\operatorname{mult}(\mu)\leq |S|$; the bound is attained when a p-th power congruence has nontrivial solutions modulo N.
For even $N>3$ with $S=\{\pm s_1,\pm s_2\}$ containing a unit, every nonzero eigenvalue has multiplicity at most 6, and this is sharp, as shown by $N=30$ with $S=\{\pm 1,\pm 3\}$, where the eigenvalue 1 has multiplicity 6.
For odd composite N, the eigenvalue $\mu_1=\sum_{s\in S}2\cos(2\pi s/N)$ is either 0 or irrational; it is 0 exactly when $|S|=2p$ and $S=A(u;N/p)$ for some u.
For square-free N, when S is a subgroup of the unit group, the eigenvalue in the m-layer is a scaled Gaussian period $\Lambda_{m,c}$; each layer contributes $\varphi(m)/|S_m|$ distinct eigenvalues, each with layer multiplicity $|S_m|$ and degree $\varphi(m)/|S_m|$ over $\mathbb{Q}$.
Two layers m and n share an eigenvalue exactly when they have the same eigenvalue set, which happens exactly when the kernels of reduction to $\gcd(m,n)$ lie in $S_m$ and $S_n$ and $\mu(r)/\varphi(r)=\mu(t)/\varphi(t)$ for $r=m/g$ and $t=n/g$.

Reading between the lines

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

The rigidity mechanism suggests a general principle beyond the paper's statement: on any cyclic group, once the weight falls below the smallest nonzero vanishing weight, one nonzero Fourier coefficient should determine the multiset; identifying that threshold for arbitrary N is a natural next step.
The zero-eigenvalue examples show that failure of rigidity is concentrated at evaluation zero, where translations of prime subgroups give identical zero spectra with different supports; this points to the kernel of the evaluation map, rather than the smallness of support, as the true obstruction to uniqueness.
The square-free layer criterion translates spectral collisions into the number-theoretic condition $\mu(r)/\varphi(r)=\mu(t)/\varphi(t)$, so the existence of eigenvalue coincidences across layers can be decided by factoring N; this makes the $N=1365$ phenomenon a member of a predictable family rather than an isolated example.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The genuinely new part is the small-weight Fourier rigidity (Theorem C) and the two-cosine fiber bound (Theorem A). The vanishing-sum classification in Theorem B is not fully proven as written.

The group-ring translation is standard, but Theorem C is a real addition: for odd N with smallest prime p, equality of one nonzero primitive Fourier coefficient forces equality of multisets up to weight 2p (symmetric case). I followed the square-free proof and the coset lifting in Section 4.2; the argument is long but structurally coherent, and the examples showing sharpness of p and 2p are useful. The Lam–Leung and Poonen–Rubinstein inputs are cited and used correctly. Theorem A is a neat byproduct: the root-of-unity proof of Wlodarski plus a finite inspection gives |R(c)| ≤ 3, with the exceptional triples listed.

The soft spot is real. In Theorem 3.2, after choosing a minimal vanishing summand Y = σ(u + C_p), the proof only handles u ∉ C_p, where Y and Y* are disjoint and can be subtracted. The self-conjugate case u ∈ C_p, where Y = Y* = C_p, is never treated; the text jumps straight to the reduced form X = eσ(C_p) + fσ(C_q). That is exactly the situation X = C_p + C_q, which vanishes and is symmetric, but subtracting C_p leaves C_q of odd weight. You need an argument that pairs the self-conjugate summand with something before subtracting. Without that, Corollary 3.3 and the zero-eigenvalue criteria in Theorems 5.2 and 5.4 are not established. This is patchable, most likely, but it is a gap in a stated if-and-only-if theorem. The stress-test note is right that Section 4 (the rigidity engine) does not rely on Theorem 3.2; I checked the dependency chain, and the multiplicity bounds in Theorem D stand independently.

Minor issues: Proposition 2.12's proof relies on a finite inspection of Wlodarski's sporadic quadruples that is summarized rather than shown in full; that is a routine check but should be spelled out. The optimality discussion in Section 5 is honest about what happens when S contains no unit.

Verdict: this deserves a serious referee, but not acceptance in current form. The author needs to close the self-conjugate gap or restrict Theorem B accordingly. If that is done, the paper is a solid contribution for number theorists working on vanishing sums of roots of unity and for spectral graph theorists interested in cyclic Cayley graphs. I would not desk-reject it; I would send it out and ask for a revision focusing on Theorem 3.2.

Referee Report

2 major / 3 minor

Summary. The paper studies structured cosine sums C_S(k)=Σ cos(2π k s_j/N) through the group-ring evaluation ev_N and vanishing sums of roots of unity. It gives a roots-of-unity proof of Włodarski's four-cosine classification, a sharp bound |R(c)|≤3 for two-cosine fibers, and block decompositions for symmetric vanishing multisets when N=p^a q^b (Theorem 3.2) and for weight 2p_1 (Theorem 3.5). The central rigidity theorem (Theorem 4.1) states that for odd N with smallest prime divisor p, equality of one nonzero primitive Fourier coefficient determines a multiset of weight ≤p uniquely, and in the symmetric case weight ≤2p uniquely. The final section applies these results to cyclic Cayley graphs: zero-eigenvalue criteria, multiplicity bounds mult(μ)≤|S| for |S|≤2p when S contains a unit, an irrationality corollary for μ_1, and, for square-free N with S a subgroup of units, a complete description of the spectrum as scaled Gaussian periods with a criterion for layer intersection.

Significance. If the proofs are completed, the paper's main contributions are solid and well motivated. The Fourier rigidity theorem is a clean, sharp result with explicit counterexamples showing the optimality of its hypotheses (Examples 4.5 and 4.13). The multiplicity bound mult(μ)≤|S| in the small-support case and its optimality construction are valuable, and the Gaussian-period description in Theorem 5.25 gives a complete spectral decomposition for square-free cyclic unit-subgroup Cayley graphs, including a precise condition for layer collisions. The paper is careful about external dependencies: it states the Lam–Leung and Poonen–Rubinstein results it uses, and the main rigidity proof in Section 4 does not rely on the problematic vanishing criterion in Theorem 3.2. I found no circularity and no fitting of free parameters. The remaining issue is a genuine but local gap in the proof of Theorem 3.2.

major comments (2)

[Section 3, proof of Theorem 3.2] The induction in the proof of Theorem 3.2 (pp. 10–11) treats only the case where the chosen minimal vanishing summand is Y=σ(u+C_p) with u∉C_p. The self-conjugate case u∈C_p, and its analogue for C_q, is never handled. The sentence 'Iterating this process for p and q, we are reduced to the case X=eσ(C_p)+fσ(C_q)' is therefore not justified: if the selected minimal summand is C_p, one cannot form a conjugate pair to subtract, and subtracting the single block C_p changes the parity of the weight, so the induction hypothesis (symmetric, even weight) no longer applies. The gap is not cosmetic, since the theorem is true for X=C_p+C_q=D_{p,q}(N) but the proof fails if C_p is the minimal summand chosen. A repair is available: first delete all non-self-conjugate translates in conjugate pairs, then use e+f≡0 mod 2 to subtract one of A(0;N/p), A(0;N/q), or D_{p,q}(N). This argument must be written into the proof. Because Corollary 3.3 and the zero-eigenvalue criteria in Theorems 5.2 and 5.4 rely on Theorem 3.2, the stated if-and-only-if results are not fully proven as written.
[Section 3, Remark 3.4 and Section 5.2, Remark 5.3] The block decomposition for N=2^a q^b is asserted in Remark 3.4 with only a reference to 'the same inductive argument', but no proof is supplied. The statement involves additional self-conjugate blocks, and handling those requires the same parity-pairing argument that is missing in the odd case of Theorem 3.2. Since Remark 5.3 converts this classification into a zero-eigenvalue criterion for even cyclic Cayley graphs, the criterion is currently conditional on an unproved assertion. Either a proof of Remark 3.4 should be included, or the statement should be explicitly marked as unproved.

minor comments (3)

[Section 1, after (1.3)] The phrase 'This is z special case' contains a typo and should read 'This is a special case'.
[Section 4.1, Remark 4.12] The sentence 'Then Theorem 2.3(3) then ε(A)=ε(B) to p,2p' is garbled. More importantly, the claimed equivalence of Propositions 4.10 and 4.11 requires an argument that symmetric disjoint vanishing multisets of equal weight at most 2p cannot have weight q, the second smallest prime divisor of M; this fact is not stated and should be justified, or the remark should be softened.
[Section 2.2, Proposition 2.12] The assertion that at most one pair in R(c) has equal reduced denominators is justified only by 'inspection' of the Włodarski list; a short table of the relevant sporadic quadruples would make this step easier to verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning found; the derivation chain is a normal dependency on published external classifications.

full rationale

No load-bearing step reduces to its own input. The paper's key inputs are Lam-Leung's minimal-vanishing-sum classification (Theorem 2.3), the Poonen-Rubinstein low-weight list (Proposition 2.9), and Lenstra's coset-slicing technique (Lemma 4.14), none of which are supplied by the present paper or fitted to its conclusions. Theorem A (Proposition 2.12) is derived from Wlodarski's theorem, which the paper reproves from the Poonen-Rubinstein list; the later use of Proposition 2.12 in Theorem 5.9 is a normal theorem-dependency chain, not a definitional equivalence. The small-weight rigidity engine is proved by induction on square-free radicals (Propositions 4.4, 4.10, 4.11) and then lifted; Remark 4.12 only notes that Propositions 4.10 and 4.11 are equivalent after their proofs and is not used as an unproved premise. The only presentation gap is in Theorem 3.2's induction when a minimal summand is self-conjugate (u in C_p): the text jumps to the reduced form, but the missing subtraction is explicitly available as A(0;N/p)=2sigma(C_p)<=X, so this is a terseness or correctness issue, not a circular one. No fitted parameter is renamed as a prediction, and the paper contains no self-citations. Score 0.

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

The central claims rest on cited classifications from Lam-Leung, Poonen-Rubinstein, and Wlodarski, plus standard cyclotomic and normal-basis facts. There are no fitted parameters and no invented entities. The new objects defined in the paper, such as the blocks A(u;N/p) and D_{p,q}(N), are explicit constructs with clear algebraic purpose, not postulated entities lacking evidence.

assumptions (6)

standard math Lam-Leung theory: for N with two distinct prime factors, the only minimal vanishing sums up to rotation are R_p and R_q; the weight set W(N) is Np_1 + ... + Np_s.
Invoked as Theorem 2.3 and used repeatedly in Sections 3 and 4; the paper does not reprove it.
standard math Poonen-Rubinstein classification: any conjugation-stable vanishing sum of eight roots of unity is one of the six forms listed in Proposition 2.9.
Used to reprove Wlodarski's four-cosine theorem, which feeds Proposition 2.12 and Theorem 5.9.
standard math Wlodarski's four-cosine vanishing classification is complete.
Reproved here from Proposition 2.9 and used as a black box in Proposition 2.12.
standard math Lam-Leung comparison theorem: for square-free N=p_1...p_s, if ν(x)≤p_1−1 then either ν(y)≥(p_1−ν(x))(p_2−1) or y≥x.
Used throughout the square-free rigidity proof in Section 4.1.
standard math For square-free m, primitive m-th roots form a normal basis of Q(ζ_m)/Q, and the sum over all primitive m-th roots is μ(m).
Used in Theorem 5.23 and Theorem 5.25 to identify Gaussian periods and layer eigenvalue intersections.
domain assumption The rigidity theorem is stated only for odd N; even N admits counterexamples.
Examples 4.5 and 4.13 show the oddness restriction is necessary; this defines the regime of the central theorem.

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Pith. "Pith review of On structured cosine sums and applications." pith.science (2026). https://pith.science/paper/ALQIQ7Y7

@misc{pith2026260720907,
  author       = {Pith},
  title        = {Pith review of: On structured cosine sums and applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ALQIQ7Y7}},
  note         = {Machine review of arXiv:2607.20907}
}

abstract

For a multiset $S$ on the cyclic group $\mathbb{Z}/N\mathbb{Z}$, we study finite sums of cosine functions of rational angles associated to $S$ by translating them as evaluations of elements in the group ring $\mathbb{Z}[\mathbb{Z}/N\mathbb{Z}]$. Using vanishing sums of roots of unity, especially the Lam-Leung theory, we obtain criteria for the vanishing of the cosine sums under some conditions, and prove a small-weight Fourier rigidity. We then apply these algebraic results to cyclic Cayley graphs, deriving the zero-eigenvalue criteria, multiplicity bounds for nonzero eigenvalues in the small-support case, and a description of the square-free case where the generating set is a subgroup of the unit group.

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Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Spectra of C ayley graphs

L \'a szl \'o Babai. Spectra of C ayley graphs. Journal of Combinatorial Theory, Series B , 27(2):180--189, 1979

work page 1979

[2] [2]

Trigonometric D iophantine equations (on vanishing sums of roots of unity)

John Conway and A Jones. Trigonometric D iophantine equations (on vanishing sums of roots of unity). Acta Arith. , 30:229--240, 1976

work page 1976

[3] [3]

On the factorization of cyclic groups

Nicolaas Govert DEBRUIJN. On the factorization of cyclic groups. Indag. Math. , 15:370--377, 1953

work page 1953

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Classification of rational angles in plane lattices

Roberto Dvornicich, Francesco Veneziano, and Umberto Zannier. Classification of rational angles in plane lattices. Bulletin of the American Mathematical Society , 59(2):191--226, 2022

work page 2022

[5] [5]

An introduction to the theory of numbers, sixth edition

Godfrey Harold Hardy and Edward Maitland Wright. An introduction to the theory of numbers, sixth edition . Oxford, 2008

work page 2008

[6] [6]

Equiangular lines with a fixed angle

Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang, and Yufei Zhao. Equiangular lines with a fixed angle. Annals of Mathematics , 194(3):729--743, 2021

work page 2021

[7] [7]

Kedlaya, Alexander Kolpakov, Bjorn Poonen, and Michael Rubinstein

Kiran S. Kedlaya, Alexander Kolpakov, Bjorn Poonen, and Michael Rubinstein. Space vectors forming rational angles, 2020. arXiv:2011.14232

arXiv 2020

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Configurations with rational angles and trigonometric D iophantine equations

M Laczkovich. Configurations with rational angles and trigonometric D iophantine equations. In Discrete and Computational Geometry: The Goodman-Pollack Festschrift , pages 571--595. Springer, 2003

work page 2003