REVIEW 2 major objections 3 minor 23 references
On the fixed volume discrepancy of the Fibonacci sets in the integral norms
T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read Fibonacci point sets reach sqrt(log) rate for periodic L_p fixed-volume discrepancy.
desk verdict Genuinely new sqrt(log) bound for L_p fixed-volume discrepancy of Fibonacci sets, with a real but minor gap in the r=1 large-box case. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the periodic $r$-smooth fixed-volume $L_p$ discrepancy $\tilde D_p^r(\xi,v)$ from Definition 1.2, which replaces the supremum over box shifts by an $L_p$ average over $z$, computed against the periodized convolution hat kernel $\tilde h_B^r$. The mechanism that carries the argument is a four-step Fourier estimate. First, the Fibonacci cubature formula is exactly zero on the hyperbolic cross $\Gamma(\gamma b_n)$, because the frequency lattice $L(n)=\{k: k_1+b_{n-1}k_2\equiv0\pmod{b_n}\}$ avoids that cross (Lemma 2.1). Second, the tail of the Fourier series is organized into dyadic shells $\rho(s)$; Lemma 2.2, imported from [16], bounds the sum $\sigma_u^r(t)$ of the shell coefficients. Third, the number of lattice points in a shell at level $t$ is at most $C 2^{2(t-t_0)}$ for $t$ beyond the critical level $t_0 \asymp \log b_n$ (inequality 2.10). Fourth, the Littlewood-Paley inequality (2.6) for $p \in [2,\infty)$ converts the $\ell^2$ sum of shell $L_p$ norms into an $L_p$ bound, producing the square root of the logarithm; for $p<2$ the result follows from $L_2$ because the underlying measure has total mass one.
What would settle it
Compute $\sigma_u^r(t)$ numerically for $r=2$, dimension $d=2$, and several $u$ with $p_r(u)\ge 2^{-t}$, and compare it with $C(\log(2^{t+1}p_r(u)))^{-1}/(2^t p_r(u))^{r/2}$; if no fixed $C$ works for arbitrarily large $t$, Lemma 2.2 is false and the proof of Theorem 1.2 collapses. A cheaper check is to evaluate $\tilde D_2^2(F_n,1/b_n)$ for moderate $n$ by direct Fourier summation: the observed rate should show the square-root-of-log factor, not the full log, for the theorem to be right.
Extended reading notes
Core claim
For the Fibonacci point set $F_n \subset [0,1)^2$ with $b_n$ points, the paper defines the periodic $r$-smooth fixed-volume $L_p$ discrepancy $\tilde D_p^r(\xi,v)$ by taking the $L_p$ norm over shifts $z$ of the difference between the integral and the sample average of the periodized smoothed box $\tilde h_B^r(\cdot - z)$ (Definition 1.2). Theorem 1.2 asserts that for every $r \in \mathbb N$ and $1 \le p < \infty$ there are constants $c,C>0$ with $\tilde D_p^r(F_n,v) \le C \sqrt{\log(b_n v)}/b_n^r$ for all $v \ge c/b_n$. The proof writes the cubature error as a Fourier series over the nonzero lattice points $k \in L(n)$; Lemma 2.1, the empty hyperbolic-cross property, kills all low frequencies, a dyadic bound imported from [16] controls the tail, and the Littlewood-Paley inequality assembles the dyadic blocks in $L_p$. The same method with the triangle inequality instead of Littlewood-Paley gives only the $p=\infty$ bound $C\log(b_n v)/b_n^r$, so the improvement is specifically a consequence of $L_p$ averaging over shifts.
Load-bearing premise
The proof relies on an unproved bound from a previous paper saying that certain sums of Fourier coefficients over dyadic frequency shells decay at a specific rate; if that bound is false, or its condition is not met for the volumes and frequencies used, the central upper bound does not follow.
Editorial extensions
If this is right
- For every $1\le p<\infty$, the periodic fixed-volume discrepancy of the Fibonacci set has the same order in the worst case over $v$ as the lower bound $m^{-r}(\log m)^{(d-1)/2}$ from [18] in dimension two: the $\sqrt{\log m}$ factor is unavoidable up to constants.
- Any box that realizes the discrepancy lower bound must have volume at least $b_n^{-1+\delta}$ for some fixed $\delta>0$; boxes of volume only $(\log b_n)^c/b_n$ cannot be the worst ones.
- The $p=\infty$ rate $m^{-r}(\log m)^{d-1}$, already known from [17], is sharp in the same sense, so the difference between the $p<\infty$ and $p=\infty$ rates is intrinsic to taking the supremum over shifts.
- Via the connection established in [16] between fixed-volume discrepancy and dispersion, the improved discrepancy bound yields the corresponding upper bound for the dispersion of Fibonacci point sets in the unit square.
- For $p<2$ no additional work is needed: the $L_p$ norm on the unit cube is dominated by the $L_2$ norm, so the $L_2$ bound automatically gives the $L_p$ bound.
Reading between the lines
- The same mechanism plausibly extends to other point sets whose error vanishes on a hyperbolic cross of size comparable to the number of points, such as rank-1 lattices with good generating vectors, giving the same $\sqrt{\log m}$ improvement over $p=\infty$; the only input needed is an empty-cross lemma like Lemma 2.1.
- The Littlewood-Paley versus triangle inequality contrast suggests a general principle: any cubature formula whose error has a lacunary Fourier spectrum will show a logarithmic gap between pointwise and $L_p$ discrepancy, with the gap exactly one square root of a logarithm.
- One could test whether the restriction to the periodic setting is essential; if the same averaging trick works for the non-periodic hat functions from Definition 1.1, the result would improve Theorem 1.1's $\log(b_n v)$ bound as well, but the paper leaves this open.
- A quantitative version might determine the optimal $\delta$ in the statement that bad boxes have volume at least $b_n^{-1+\delta}$ by matching the lower-bound constructions of [18] against the upper-bound proof.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper studies the periodic r-smooth fixed volume L_p discrepancy of the Fibonacci point sets in dimension two. The authors introduce a discrepancy functional obtained by averaging over shifts of a smooth hat function whose support is a box of prescribed volume v, and they prove (Theorem 1.2) that for every r in N and 1 <= p < infinity there are constants c,C > 0 such that for v >= c/b_n the discrepancy of the Fibonacci set F_n is at most C sqrt(log(b_n v)) / b_n^r. A weaker sup-norm bound with an additional logarithm is given in Theorem 1.3 for r >= 2. The proofs combine the Fibonacci lattice's exact Fourier cancellation on hyperbolic crosses with Littlewood-Paley decomposition and a dyadic sum estimate imported from the authors' earlier work [16].
Significance. If the theorem is correct, it gives a near-optimal, up to the square root of a logarithm, L_p fixed-volume discrepancy bound for Fibonacci sets, improving on the p = infinity log-rate and supporting the paper's interpretation that 'bad boxes' cannot be too small. The use of shift-averaging over hat functions is a genuine new ingredient, and the main derivation is explicit and largely self-contained. However, two load-bearing issues in the written proof must be repaired before the result is fully established: the stated hypothesis of Lemma 2.2 does not cover all boxes relevant to the r = 1 case, and the counting bound in (2.10) appears inconsistent with the subsequent series estimate.
major comments (2)
- [§2, equation (2.10) and the following estimate] Equation (2.5) is asserted for all r >= 1 and all t with v >= r^d 2^{-t+1}, but it is derived from Lemma 2.2, which is stated only for u in (0,1/2]^d. For r = 1, the boxes B subset [0,1)^2 allowed in Definition 1.2 may have side lengths u_j > 1/2, for example when the volume is close to 1, so Lemma 2.2 does not apply to those boxes. Since Theorem 1.2 includes r = 1 and the proof's final series requires only r > 1 - 1/p, the r = 1 case is genuinely in scope. The manuscript must supply an extension of Lemma 2.2 to u in (0,1]^d, or an alternative argument for coordinates u_j in [1/2,1], and must state the resulting condition on p_r(u); as written, the central upper bound for r = 1 is not proved.
- [§2] The counting bound in (2.10) is stated as #(rho(s) cap L(n)) <= C 2^{2t-t0}, but the next displayed inequality uses the factor 2^{2(t-t0)(1-1/p)}, which corresponds to # <= C 2^{t-t0} rather than # <= C 2^{2t-t0}. If (2.10) were used as printed, the exponent in the series would be 2(2t-t0)(1-1/p) - 2rt; for the central case r = 1, p = 2 this gives a term of order 2^{-t0} log(2^t v), whose sum over t diverges. The proof therefore requires either correcting (2.10) to the sharper per-block count # <= C 2^{t-t0}, with a justification from the Fibonacci lattice structure, or providing a two-regime argument. This issue is load-bearing for Theorem 1.2 and also affects the one-sentence proof of Theorem 1.3.
minor comments (3)
- [§2] The derivation of (2.5) applies Lemma 2.2 with smoothness parameter 2r rather than r, because H_B^r(s)^2 contains the square of the min-factor; this is not stated and should be explained for readability.
- [Text near (2.9)] There is a typo: 'Parselval's identity' should read 'Parseval's identity'.
- [Theorem 1.3 proof] The proof of Theorem 1.3 is only sketched; the claim that 'we need r > 1 for the last series' is not demonstrated in the text and should be expanded, especially in view of the counting issue in (2.10).
Circularity Check
No circularity: the upper bound in Theorem 1.2 follows from a concrete Fourier and Littlewood-Paley argument; the imported Lemma 2.2 is a technical input, not an equivalent restatement of the result.
full rationale
The central claim, Theorem 1.2, is proved directly from Fourier coefficient bounds, the hyperbolic-cross separation property of the Fibonacci lattice (Lemma 2.1), a Littlewood-Paley norm inequality, and the dyadic-sum estimate Lemma 2.2. Lemma 2.2 is imported from the authors' earlier work [16, Lemma 6.1], so there is a self-citation that is load-bearing in the high-frequency tail estimate. However, this is not circularity: Lemma 2.2 is a standalone dyadic-sum bound with explicit hypotheses, and it is not equivalent to the discrepancy upper bound being derived, nor is it fitted to the Fibonacci point set or to the specific discrepancy definition. The lower bounds cited from [17] and [18] are also self-citations, but they are used only to discuss sharpness, not to prove the upper bound. The skeptical concern that Lemma 2.2 is stated for u in (0,1/2]^d while the proof applies it in the r=1 regime with u_j possibly greater than 1/2 is a possible correctness gap in the written proof, not a circularity; the missing step would be an extension of a dyadic estimate, not a reduction of the theorem to its own assumptions. Therefore no circular step is exhibited, and the derivation is self-contained apart from the cited technical lemma.
Assumptions & free parameters
assumptions (4)
- standard math Littlewood-Paley inequality (2.6) for p in [2,∞): ||f||_p ≤ C(d,p) (sum_{s} ||δ_s(f)||_p^2)^{1/2}
- standard math Fibonacci lattice Fourier coefficient property (2.2) and the hyperbolic cross emptiness Lemma 2.1
- standard math Dyadic sum estimate Lemma 2.2, imported from [16, Lemma 6.1]
- standard math Monotonicity of L_p norms on probability spaces
Cite this review
Pith. "Pith review of On the fixed volume discrepancy of the Fibonacci sets in the integral norms." pith.science (2026). https://pith.science/paper/YLHDFRY4
@misc{pith2026190804658,
author = {Pith},
title = {Pith review of: On the fixed volume discrepancy of the Fibonacci sets in the integral norms},
year = {2026},
howpublished = {\url{https://pith.science/paper/YLHDFRY4}},
note = {Machine review of arXiv:1908.04658}
}
abstract
This paper is devoted to the study of a discrepancy-type characteristic -- the fixed volume discrepancy -- of the Fibonacci point set in the unit square. It was observed recently that this new characteristic allows us to obtain optimal rate of dispersion from numerical integration results. This observation motivates us to thoroughly study this new version of discrepancy, which seems to be interesting by itself. The new ingredient of this paper is the use of the average over the shifts of hat functions instead of taking the supremum over the shifts. We show that this change in the setting results in an improvement of the upper bound for the smooth fixed volume discrepancy, similarly to the well-known results for the usual $L_p$-discrepancy. Interestingly, this shows that ``bad boxes'' for the usual discrepancy cannot be ``too small''. The known results on smooth discrepancy show that the obtained bounds cannot be improved in a certain sense.
Reference graph
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