pith. sign in

arxiv: 2511.22802 · v2 · submitted 2025-11-27 · 🧮 math.DS · math.NT

Birkhoff Measures, Birkhoff Sums, and Discrepancies

D. Ralston , F.M. Tangerman , J.J.P. Veerman , H. Wu This is my paper

Pith reviewed 2026-05-17 04:28 UTC · model grok-4.3

classification 🧮 math.DS math.NT
keywords Birkhoff sumsdiscrepancyirrational rotationscontinued fractionsuniform distributionergodic theorydynamical systems
0
0 comments X

The pith

The length of the support of the Birkhoff measure can be expressed in terms of the discrepancy.

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

The paper examines sequences on the circle generated by repeated additions of a fixed irrational rotation ρ. It defines the Birkhoff sum as the cumulative sum of deviations of fractional parts from one half, yielding a piecewise-linear map with exactly n branches. The Birkhoff measure is then constructed as the normalized density of preimages under this map. The central result establishes that the length of the interval on which this density is positive can be written directly as a function of the discrepancy of the original sequence. The work further shows that the graph of the measure approximates an isosceles trapezoid whenever n is a continued-fraction denominator of ρ and supplies short new proofs of two classical computation results.

Core claim

We prove that the length of the support of the Birkhoff measure ν(ρ,n,z) dz can be expressed in terms of the discrepancy. We also show that if n is a continued fraction denominator of ρ, then the graph of ν(ρ,n,z) is an approximate isosceles trapezoid. New brief proofs are given for results of Ramshaw and of Kuipers-Niederreiter that enable efficient computation of Birkhoff sums and discrepancies.

What carries the argument

The Birkhoff measure ν(ρ,n,z), the normalized count of preimages of each z under the n-branched piecewise-linear Birkhoff sum map S(ρ,n,x), whose support length is shown to equal a function of the discrepancy.

If this is right

  • Discrepancy of the rotation sequence can be recovered from the support length of the associated Birkhoff measure.
  • When n is a continued-fraction denominator the graph of ν(ρ,n,z) approximates an isosceles trapezoid.
  • Birkhoff sums and discrepancies admit efficient computation via the new short proofs of the classical results.
  • The variation of the measure graph with n is controlled by the quality of distribution of the sequence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The support-length formula may extend to other ergodic sums on the circle or to toral rotations in higher dimensions.
  • Numerical checks for successively better rational approximations to ρ could quantify how rapidly the trapezoid shape emerges.
  • The relation supplies a concrete bridge between uniform-distribution quantities and the range of partial-sum functions.

Load-bearing premise

The assumption that ρ is irrational ensures density and equidistribution while the Birkhoff sum map remains piecewise linear with exactly n branches of slope n.

What would settle it

For golden-ratio ρ and n equal to the third continued-fraction denominator, compute the discrepancy of the first n points and the length of the interval where ν(ρ,n,z) is positive; check whether the two quantities satisfy the claimed functional relation.

Figures

Figures reproduced from arXiv: 2511.22802 by D. Ralston, F.M. Tangerman, H. Wu, J.J.P. Veerman.

Figure 1.1
Figure 1.1. Figure 1.1: The construction of the Birkhoff measure for ρ equal to the golden mean with n = 13. The main result in section 3 is that the length of the support of the Birkhoff density ν(ρ, n, z) is simply the discrepancy of the sequence {iρ} n i=1 multiplied by n. The main result of section 4 occurs when pn/qn is a continued fraction convergent of ρ. We show that then the graph of ν(ρ, qn, z) is an approximate isosc… view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: For ρ the golden mean, we show the range of S(ρ, n, x), indicated by the min/max values for the sum S over x ∈ [0, 1) with increasing value of n. The plot also shows S(ρ, n, x0) for two initial conditions indicated, x0 = 0 and x0 = 1/ √ 5. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_1_2.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: The sum of seven translates of the Birkhoff measure associated with e − 2 and n = 2024. Theorem 2.2 (Tiling Property) For every fixed irrational ρ and positive integer n, the density ν(ρ, n, z) satisfies: i) For every z ∈ [0, 1): P i∈Z ν(ρ, n, z + i) = 1, ii) The support of the measure associated with the density ν(ρ, n, z) is a closed and bounded interval. Proof. To prove (i), note that X i∈Z ν(ρ, n, z … view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Sketch of the branches of S with d = 0 in red and with d > 0 in blue. Here, Sj is the third branch. The lengths of the segments indicated by A, B, and C, are, respectively, (q+1)d 2 , i+d q , and (q+1−2i+)d 2 . Proof. Fix p, q, and ρ. By definition of ν, its value at any point z equals the number of solutions x of S(x) = z divided by q. Since S has q branches, then ν(ρ, q, z) 6= 1 implies that ν(ρ, q, z)… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Illustration of how changing the reduced residue in ρ simplifies the construction of ν(ρ, q, z). We first construct ν(e − 2, 32, z). knowing that 23/32 is an odd approximant of e − 2, we then construct ν(e − 2 − 22/32, 32, z). Note that there is 1 longer interval and 31 shorter ones. Corollary 4.7 (The Trapezoid Theorem) If pn/qn is a continued fraction convergent of ρ, then ν(ρ, qn, z) is an isosceles t… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Adding the two sequences of Lemma 5.1 and splitting the result into three parts. For brevity, {(i − k)ρ} − 1 2 is indicated by its index i − k, etc. Proof. Add the two sequences and split up the result into three parts as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p011_5_1.png] view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: The Birkhoff sums of the origin through times q2 = 67, q3 = 140, q4 = 207, q5 = 1388 for ρ with repeated continued fraction [6, 11, 2, 1,]. The sums through the previous qn−1 are in a shaded rectangle and horizontal axis ticks give multiples of this length. The parabola-like structure formed by the an affine translations is apparent when an is large (e.g. for q2 and q5) but less clear when an is small (e… view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: Some Birkhoff measures for ρ = e − 2. [3] Y. Bugeaud, Distribution Modulo One and Diophantine Approximation, Cambridge University Press, 2012. [4] J. P. Conze , S. Le Borgne. On the CLT for rotations and BV functions, Annales Math´ematiques Blaise Pascal. 2022, 29(1): 51-97 17 [PITH_FULL_IMAGE:figures/full_fig_p017_7_1.png] view at source ↗
read the original abstract

We study the distribution of a sequence of points in the circle generated by rotations by a fixed irrational number $\rho$ with initial condition $x_0$, that is: $\{x_0+i\rho\}_{i=1}^n$. The \emph{discrepancy} as defined by Pisot and Van Der Corput \cite{VdCP}, quantifies how evenly distributed such a sequence is. Consider the ergodic or Birkhoff sum of mean zero $S(\rho,n,x):=\sum_{i=1}^{n} (\{x+i\rho\}-1/2)$, where $\{\cdot\}$ denotes the fractional part. This is a piecewise-linear map in the variable $x$ with $n$ branches, each with slope $n$. For fixed $n$ and $\rho$, let $\nu(\rho,n,z)$ be the number of pre-images of $S(\rho,n,x)=z$ divided by $n$. Then $\nu(\rho,n,z)$ is a probability density. We call the associated measures Birkhoff measures. We investigate how the graph of $\nu(\rho,n,z)$ varies with $n$. We prove that the length of the support of the Birkhoff measure $\nu(\rho,n,z)dz$ can be expressed in terms of the discrepancy. We also show that if $n$ is a continued fraction denominator of $\rho$, then the graph of $\nu(\rho,n,z)$ an approximate isosceles trapezoid. We also give new, brief, proofs of two classical results, one by Ramshaw \cite{Ramshaw} and one found by Kuipers-Niederreiter \cite{KN}. These results allow efficient computation of both Birkhoff sums and discrepancies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript studies Birkhoff sums S(ρ,n,x) = ∑_{i=1}^n ({x + iρ} - 1/2) for irrational ρ, which are piecewise-linear maps with exactly n branches each of slope n. It defines the Birkhoff measure with density ν(ρ,n,z) as the normalized number of preimages under S, proves that the length of the support of ν(ρ,n,z) dz equals an explicit function of the discrepancy (in the Pisot–Van der Corput sense), shows that the graph approximates an isosceles trapezoid when n is a continued-fraction denominator, and supplies new brief proofs of two classical results (Ramshaw and Kuipers–Niederreiter) that enable efficient computation of sums and discrepancies.

Significance. If the central claims hold, the work supplies a direct geometric relation between discrepancy and the range of the Birkhoff sum, together with a concrete shape description for well-approximating denominators. These are parameter-free derivations from the standard definition of discrepancy and the integral representation of S, which strengthens the contribution to uniform-distribution and ergodic theory. The explicit support-length formula and the trapezoid observation are falsifiable and potentially useful for computation.

minor comments (2)
  1. [Support-length derivation] The claim that the image of S is the full interval [min S, max S] with no gaps (used for the support-length result) would benefit from an explicit verification that the chained linear segments of rise n l_j connected by drops of 1 cover the interval continuously; this appears in the geometry argument but could be isolated as a short lemma.
  2. [Trapezoid result] The trapezoid-shape statement for continued-fraction denominators is stated as 'approximate'; a quantitative error bound in terms of the next continued-fraction coefficient would make the claim sharper and easier to verify numerically.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the positive assessment of its contributions to uniform distribution theory and ergodic theory, and the recommendation to accept. We are pleased that the explicit support-length formula and the trapezoid approximation for continued-fraction denominators are viewed as falsifiable and potentially useful.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from definitions

full rationale

The paper begins with the standard definition of the Birkhoff sum S(ρ,n,x) as the sum of fractional-part deviations and the Pisot-Van der Corput discrepancy. It derives the support length of ν(ρ,n,z) as n D_n from the piecewise-linear geometry with n branches of slope n and the integral representation S = ∫ (N(s,x) - n s) ds. The image interval [min S, max S] is shown to be gap-free by explicit chaining of rises and drops. These steps use only the arithmetic-progression structure for irrational ρ and produce the claimed relation without reducing to a fitted parameter or self-referential loop. Citations to Ramshaw and Kuipers-Niederreiter support corollaries only and are not load-bearing for the central support-length result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the standard assumption that ρ is irrational together with the piecewise-linear structure of the Birkhoff sum; the Birkhoff measure itself is defined from the preimage count rather than postulated as an independent physical object.

axioms (1)
  • domain assumption ρ is irrational
    Invoked to guarantee that the orbit is dense and the sequence is equidistributed on the circle.
invented entities (1)
  • Birkhoff measure ν(ρ,n,z) no independent evidence
    purpose: Probability density obtained by normalizing the number of preimages of the Birkhoff sum S(ρ,n,x)=z.
    Defined directly from the preimage count of the piecewise-linear map; no independent physical evidence is claimed.

pith-pipeline@v0.9.0 · 5633 in / 1402 out tokens · 32348 ms · 2026-05-17T04:28:49.447255+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    Van Aardenne-Ehrenfest, Proof of the Impossibility of a Just Distribution of an Infinite Sequence of Points Over an Interval , Indag

    T. Van Aardenne-Ehrenfest, Proof of the Impossibility of a Just Distribution of an Infinite Sequence of Points Over an Interval , Indag. Math. 7, 71-76, 1945. And: On the Impossibility of a Just Distribution , indag. Math. 11, 264-269, 1949

    work page 1945

  2. [2]

    Baxa, Comparing the distribution of (n α )-sequences,Arithmetica, XCIV.4 (2000)

    C. Baxa, Comparing the distribution of (n α )-sequences,Arithmetica, XCIV.4 (2000). 16 -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 n=142 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 n=213 -2 -1 0 1 2 0 0.1 0.2 0.3 0.4 0.5 0.6 n=284 -2 -1 0 1 2 0 0.1 0.2 0.3 0.4 0.5 0.6 n=334 0.1 0.2 0.3 0.4 0.48 0.5 0.52 0.54 0.56 0.58 n=334 detail -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8...

    work page 2000

  3. [3]

    Bugeaud, Distribution Modulo One and Diophantine Approximation , Cambridge University Press, 2012

    Y. Bugeaud, Distribution Modulo One and Diophantine Approximation , Cambridge University Press, 2012

    work page 2012

  4. [4]

    J. P. Conze , S. Le Borgne. On the CLT for rotations and BV functions , Annales Math´ ematiques Blaise Pascal. 2022, 29(1): 51-97 17

    work page 2022

  5. [5]

    J. G. Van der Corput, C. Pisot, Sur la Discr´ epance Modulo Un, Indag. Math. 1, 143-153, 184-195, 260-269, 1939

    work page 1939

  6. [6]

    Dupain and V

    Y. Dupain and V. T. Sos, On the discrepancy of (n α ) sequences , in: Topics in Classical Number Theory, Vol. 1, Colloq. Math. Soc. Janos Bolyai 34, G. Halasz (ed.), North-Holland, Amsterdam, 1984, 355–387

    work page 1984

  7. [7]

    Drmota, R

    M. Drmota, R. F. Tichy, Sequences, Discrepancies and Applications, Springer Lecture Notes in Math- ematics 1651, Springer, Springer, 1997

    work page 1997

  8. [8]

    E. Hecke. ¨Uber analytische Funktionen und die Verteilung von Zahlen m od. Eins. Abhandlungen aus dem Mathematischen Seminar der Universitaet Hamburg, 1(1) : 54–76, 1922

    work page 1922

  9. [9]

    Hlawka, Theorie der Gleichverteilung , Bibliographisches Inst., Mannheim, 1979

    E. Hlawka, Theorie der Gleichverteilung , Bibliographisches Inst., Mannheim, 1979

    work page 1979

  10. [10]

    Kesten, On a conjecture of Erd¨ os and Sz¨ usz related to uniform distribution mod 1 , Acta Arith

    H. Kesten, On a conjecture of Erd¨ os and Sz¨ usz related to uniform distribution mod 1 , Acta Arith. 12 (1966/1967) 193-212

    work page 1966

  11. [11]

    H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods , CBMS-NSF Regional Conference Series in Applied Mathematics, 63, Society for I ndustrial and Applied Mathematics, Philadelphia, PA, 1992

    work page 1992

  12. [12]

    Kravitz, Computational study of irrational rotations via exact disco ntinuity tracking , arXiv:2511.13879v1, 2025

    H. Kravitz, Computational study of irrational rotations via exact disco ntinuity tracking , arXiv:2511.13879v1, 2025

    work page arXiv 2025

  13. [13]

    Kuipers, H

    L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences , John Wiley & Sons, New york, 1974

    work page 1974

  14. [14]

    Ramshaw, On the discrepancy of the sequence formed by the multiples of an irrational number , J

    L. Ramshaw, On the discrepancy of the sequence formed by the multiples of an irrational number , J. Number Theory 13 (1981), 138–175

    work page 1981

  15. [15]

    Ralston, title TBD ,2025

    D. Ralston, title TBD ,2025

    work page 2025

  16. [16]

    Schoissengeier, On the Discrepancy of (n α II), Journal of Number Theory,24, 54-64 ( 1986)

    J. Schoissengeier, On the Discrepancy of (n α II), Journal of Number Theory,24, 54-64 ( 1986)

    work page 1986

  17. [17]

    W. H. Schmidt, Irregularities of Distribution. VII , Acta Arith. 21, 45-50, 1972

    work page 1972

  18. [18]

    Strauch, S

    O. Strauch, S. Porubsk´ y, Distribution of Sequences: A Sampler , Electronic revised version by Peter Lang, 2018, https://www.mat.savba.sk/musav/O_Strauch-S_Porubsky_Distribution_of_ Sequences_A_Sampler-20180118.pdf

    work page 2018

  19. [19]

    J. J. P. Veerman, Numbers from all Angles , In Press, Springer Nature 2025. 18

    work page 2025