REVIEW 4 minor 5 references
A problem concerning Riemann sums
T0 review · 0 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash
Pith's one-line read For merely continuous functions, the differences of successive Riemann sums need not converge, answering an open problem in the negative; the proof uses Brownian motion.
desk verdict Short, self-contained negative answer to an open problem about Riemann sums; the probabilistic argument is sound, with only a typographical blemish in the moment calculation. 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 load-bearing object is the standard Wiener process, or Brownian motion, $W=(W(u))_{u\in[0,1]}$: a random continuous function whose covariance identity $E W(u)W(v)=u\wedge v$ turns lattice sums of its values into exact polynomial expressions in $n$. The proof uses this identity to compute $E x_n^2$, $E x_nx_{n+1}$, and the four cross-moments displayed in Lemma 4, obtaining $E(y_{4s}-y_{2s})^2\to 1/4$. Since a sequence of differences that converged almost surely to $0$ would converge in distribution to $0$, this non-degenerate normal limit is what forces the existence of a continuous counterexample.
What would settle it
Recompute the four cross-moment sums in Lemma 4 and check whether $E(y_{4s}-y_{2s})^2$ tends to $1/4$; if the limit is actually $0$, the subsequence differences could vanish and the contradiction would disappear. A concrete numerical check is to simulate Brownian paths on the grid $\{0,1/(4s),\ldots,1\}$, compute $y_{4s}-y_{2s}$ for large $s$, and estimate its variance: values approaching $1/4$ confirm the central calculation, while values approaching $0$ would refute it.
Extended reading notes
Core claim
The central discovery is that continuity is not a sufficient condition for convergence of the differences of consecutive Riemann sums. The paper answers the open problem in the negative by proving existence of continuous $f:[0,1]\to\mathbb{R}$ with $\lim_{n\to\infty} y_n$ nonexistent, while also proving $y_n\to\int_0^1 f(x)\,dx$ whenever $f$ is absolutely continuous and that any existing limit must equal that integral. The existence proof is probabilistic: choosing $f$ to be a Brownian path gives an almost surely continuous function for which $y_{4s}-y_{2s}$ converges in distribution to $N(0,1/4)$, which cannot happen if $y_n\to\int f$ almost surely. Thus continuity of a function is genuinely not enough; some stronger smoothness, such as absolute continuity, is needed.
Load-bearing premise
The proof depends on the standard theorem that a Brownian motion exists with continuous sample paths and covariance $E W(u)W(v)=u\wedge v$; if that existence theorem were not available, the construction of the random continuous counterexample would collapse.
Editorial extensions
If this is right
- The open problem is answered in the negative: continuity alone does not force $y_n$ to have a limit.
- For absolutely continuous $f$, the limit always exists and equals $\int_0^1 f(x)\,dx$, and for any continuous $f$ whose sequence $y_n$ converges, that integral is the only possible limit.
- The proof is non-constructive, so explicitly exhibiting a continuous function with $\lim_n y_n$ nonexistent remains open.
- Any sufficient condition for convergence of $y_n$ must be strictly stronger than continuity; absolute continuity is one such condition.
Reading between the lines
- Editorial inference: the proof actually indicates a stronger statement: under the measure on continuous functions induced by Brownian motion, almost every function is a counterexample, with convergence of $y_n$ occurring only on a set of paths of probability zero.
- Editorial inference: the same covariance computations could be repeated with other subsequence ratios, such as $y_{rs}-y_s$, likely producing normal limits whose variance depends on $r$; that would show the oscillation is not confined to one pair of subsequences.
- Editorial inference: since Brownian paths are nowhere differentiable and fail Hölder continuity of every order at least $1/2$, the smoothness threshold for convergence may lie at some fractional regularity; a testable conjecture is that Hölder continuity of order greater than $1/2$ guarantees convergence.
- Editorial inference: a deterministic counterexample might be constructed from a Weierstrass-type trigonometric series with carefully chosen frequencies, since the probabilistic argument proves existence but gives no explicit function.
Formalized claims in Lean
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Claim #1: The central discovery is that continuity is not a sufficient condition for convergence of the differences of consecutive Riemann sums. The paper answers the open problem in the negative by proving existence of continuous $f:[0,1]\to\mathbb{R}$ with $\lim_{n\to\infty} y_n$ nonexistent, while also proving $y_n\to\int_0^1 f(x)\,dx$ whenever $f$ is absolutely continuous and that any existing limit mus
/-- @claim 1 The central discovery is that continuity is not a sufficient condition for convergence of the differences of consecutive Riemann sums. The paper answers the open problem in the negative by proving existence of continuous $f:[0,1]\to\mathbb{R}$ with $\lim_{n\to\infty} y_n$ nonexistent, while also proving $y_n\to\int_0^1 f(x)\,dx$ whenever $f$ is absolutely continuous and that any existing limit mus -/ def central_claim : Prop :=
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the sequence y_n = x_{n+1} - x_n, where x_n is the sum of f(k/n) for k=1,...,n-1 and f is continuous on [0,1]. Proposition 1 shows that if f is absolutely continuous, then y_n converges to the integral of f. Proposition 2 shows that whenever y_n converges, its limit must be that integral. The main result, Theorem 3, answers Furdui's Problem 1.32(b) by showing that there exist continuous functions for which y_n does not converge. The proof is probabilistic: taking f to be a Brownian path W, Lemma 4 computes the limiting variance of y_{4s} - y_{2s} as 1/4, which contradicts convergence of y_n, since convergence would force this difference to vanish in distribution. The paper also proves Propositions 1 and 2, the latter by the Stolz-Cesaro theorem.
Significance. If correct, the paper settles an open problem in the negative, showing that continuity alone does not guarantee convergence of the Riemann-sum increments y_n. The result is interesting and concise, and the probabilistic method is well chosen: it converts a deterministic existence question into a variance computation for a Gaussian process. The proof is essentially self-contained, relying only on standard Brownian motion facts, covariance calculations, L1 density of continuous functions, and the Stolz-Cesaro theorem. The explicit covariance computations in Lemma 4 are a strength, and the distinction between Propositions 1 and 2 is clean and illuminating.
minor comments (4)
- [Proof of Lemma 4] In the displayed computation of E x_{4s+1}x_{2s+1}, the upper limit of the second inner sum is printed as 4s-1, but the stated final value (32s^3+14s^2)/(12s+3) corresponds to the upper limit 4s; this is a typographical error only and does not affect the mathematics.
- [Proof of Lemma 4] The phrase 'direct calculations' covers four nontrivial double-sum evaluations; since these identities are the computational heart of the lemma, expanding at least one of them, or stating that they can be verified by a computer algebra system, would improve verifiability and readability.
- [Proof of Theorem 3] The statement 'without loss of generality we may assume that all realizations of W are everywhere continuous' is slightly imprecise: standard Brownian motion has a continuous modification, and one should formally work with that modification. The intended meaning is clear, but a one-sentence clarification would remove any ambiguity.
- [Proposition 1] Equation (10) identifies I_n(g) with f(n/(n+1)) - f(0); the reader must recall the fundamental theorem for absolutely continuous functions. Adding a brief parenthetical would make the step more transparent, though it is not logically problematic.
Circularity Check
No significant circularity: the proof is self-contained against standard external results, with no fitted inputs or self-citation chains.
full rationale
The proof of Theorem 3 is non-circular. Its key probabilistic ingredient is the standard Wiener process with almost surely continuous paths and covariance E W(u)W(v)=u∧v; this is imported from the external textbook [4], and it is used only to produce a continuous function with explicitly computable Gaussian finite-dimensional distributions. Lemma 4 is then proved by direct double-sum calculations from the covariance formula; no parameter is fitted, no data subset is predicted, and no claim of the present paper is assumed. The contradiction argument has the correct logical structure: assuming all continuous f have convergent y_n forces y_{4s}-y_{2s}->0 for f=W almost surely, whereas Lemma 4 computes its distribution limit as N(0,1/4), giving the needed contradiction. Proposition 1 is a standard L1-density approximation argument, and Proposition 2 is a direct application of the Stolz-Cesaro theorem. The only apparent issue is typographical, concerning an upper summation limit in an intermediate display in the proof of Lemma 4, and it does not affect the limiting variance. No self-citation is used, and no circular step is present.
Assumptions & free parameters
assumptions (7)
- standard math Existence of a standard Wiener process (Brownian motion) on [0,1] with continuous sample paths.
- standard math Covariance of Brownian motion: E W(u)W(v) = u∧v.
- standard math Density of continuous functions in L^1[0,1].
- standard math Absolute continuity implies f(x)=f(0)+∫_0^x g(u)du for some g∈L^1.
- standard math Stolz-Cesaro theorem.
- standard math For centered Gaussian random variables, convergence of variances implies convergence in distribution.
- standard math If a sequence of random variables converges almost surely to 0, then it converges in distribution to 0.
Cite this review
Pith. "Pith review of A problem concerning Riemann sums." pith.science (2026). https://pith.science/paper/TKOSTBCN
@misc{pith2026200913719,
author = {Pith},
title = {Pith review of: A problem concerning Riemann sums},
year = {2026},
howpublished = {\url{https://pith.science/paper/TKOSTBCN}},
note = {Machine review of arXiv:2009.13719}
}
read the original abstract
An open problem concerning Riemann sums, posed by O. Furdui, is considered.
Reference graph
Works this paper leans on
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[1]
V . I. Bogachev. Measure theory. Vol. I, II . Springer-V erlag, Berlin, 2007
2007
- [2]
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[3]
O. Furdui. Limits, series, and fractional part integrals . Problem Books in Mathematics. Springer, New Y ork, 2013. Problems in mathematical analysis
work page 2013
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[4]
P . M¨ orters and Y . Peres.Brownian motion, volume 30 of Cambridge Series in Statistical and Probabilis- tic Mathematics. Cambridge University Press, Cambridge, 2010. With an appe ndix by Oded Schramm and Wendelin Werner
work page 2010
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[5]
O. Stolz. V orlesungen ¨ uber allgemeine Arithmetik: nach den NeuerenAnsichten. Teubners, 1885. Iosif Pinelis, Department of Mathematical Sciences Michigan Technological University Houghton, Michigan 49931, USA e-mail: ipinelis@mtu.edu Corresponding Author: Iosif Pinelis 5
Reviewed August 27, 2026 · model on record in the stance chip above.
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