pith. sign in

arxiv: 2605.31515 · v1 · pith:AWZ5F6UKnew · submitted 2026-05-29 · 🧮 math.NT

On the asymptotic average diameter of blocks of uniformly distributed sequences and related results

Sebastian Heintze , Wolfgang Trutschnig This is my paper

Pith reviewed 2026-06-28 20:49 UTC · model grok-4.3

classification 🧮 math.NT
keywords uniformly distributed sequencesasymptotic average diameterpairwise distancesblocks of pointsd-stochastic measuressharp upper boundsoptimization problem
0
0 comments X

The pith

Optimization over d-stochastic measures yields sharp upper bounds on asymptotic average pairwise distances in blocks of uniformly distributed sequences.

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

The paper derives sharp upper bounds for the asymptotic averages of maximal, minimal, and total pairwise distances within every block of d consecutive points drawn from a uniformly distributed sequence on the unit interval. It proceeds by enlarging the search space to include all d-stochastic measures, solving the resulting compact optimization problem, and verifying that the attained maxima are also the least upper bounds for the original sequence problem. A reader would care because the bounds quantify the strongest possible local clustering that can still occur inside finite segments of any such sequence, extending earlier results limited to consecutive-point distances.

Core claim

The supremum of the asymptotic average of each of the three distance aggregations, taken over all uniformly distributed sequences, equals the maximum value of the corresponding functional on the compact family of d-stochastic measures, and explicit sharp bounds are obtained by solving that optimization problem.

What carries the argument

The family of d-stochastic measures, which enlarges the problem to a compact optimization whose solutions are then shown to be attainable as limsups for actual uniformly distributed sequences.

If this is right

  • The same sharp bounds apply simultaneously to maximal pairwise distance, minimal pairwise distance, and total pairwise distance within each block.
  • Explicit numerical values for the bounds follow directly from the solution of the measure optimization problem.
  • The bounds remain valid when the sequence is restricted to any interval of the unit interval.
  • The method recovers and generalizes prior bounds for the special case of consecutive points (d=2).

Where Pith is reading between the lines

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

  • The relaxation step may be reusable for other additive functionals defined on finite blocks of sequences.
  • The explicit bounds could be used to test candidate low-discrepancy sequences for local clustering violations.
  • Similar measure-theoretic enlargements might handle blocks in higher-dimensional or non-Euclidean spaces.

Load-bearing premise

The supremum attained by the relaxed optimization over d-stochastic measures equals the limsup of the corresponding quantity for uniformly distributed sequences.

What would settle it

A uniformly distributed sequence whose limsup of one of the three average distance quantities strictly exceeds the value obtained from the d-stochastic measure optimization.

Figures

Figures reproduced from arXiv: 2605.31515 by Sebastian Heintze, Wolfgang Trutschnig.

Figure 1
Figure 1. Figure 1: Example of a (checkerboard representation of a) per￾mutation of [N] with N = 7. Equation (7) involves exactly d permutations. When overlaying them in one picture, each row can contain at most d different black squares (note that they could overlap and therefore we have ‘at most’ instead of ‘exactly’). We are interested in maximizing the sum over the distances of the highest and lowest black block of each c… view at source ↗
Figure 2
Figure 2. Figure 2: Exemplary aggregation of d permutations of [N] for the case d = 3, N = 7 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Establishing exactly d squares in the highest row. only increases the sum of interest (7). Next we take a look at the second to highest row. If there are less than d different black columns in this row, then we move the highest black block of some column(s), where it currently is located in a lower row, up to the currently handled row until there are exactly d different black columns in the currently handl… view at source ↗
Figure 4
Figure 4. Figure 4: Exactly d = 3 black squares in the second row. Again this procedure of moving black blocks only increases the sum of distances between the highest and lowest black block in each column. We proceed in this row￾wise manner until there are no highest black blocks in a row below the currently handled one left (see [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Finishing the rearrangment of the highest black blocks. or until there is no lowest black block in a higher row remaining which could be moved down. The procedure ends when there are no lowest black blocks in a row above the currently handled one left. As before, this procedure of moving black blocks only increases the sum of distances between the highest and lowest black block in each column (see [PITH_F… view at source ↗
Figure 6
Figure 6. Figure 6: Optimize the lowest black blocks. This procedure yields the situation in which the row containing the lowest one of the highest black blocks of each column is not below the row containing the highest one of the lowest black blocks of each column; depending on d and N, it is either above or they are equal. For seeing this, note that the highest (or lowest, respectively) blocks of the columns, after the befo… view at source ↗
Figure 7
Figure 7. Figure 7: Separation of highest and lowest black blocks. Starting with a constellation of the afore-mentioned type we now calculate the sum of distances between the highest and lowest black block in each column, which yields an upper bound for the sum in expression (7). If in this configuration we exchange the highest black blocks from two columns, then the sum of distances between the highest and lowest black block… view at source ↗
Figure 8
Figure 8. Figure 8: Reordering of the highest and lowest black blocks. In this situation, however, it is now straightforward to compute the sum S of distances, which yields an upper bound as desired. Writing N = dK + R with 0 ≤ R < d, we get X N n=1 max i,j∈[d], i̸=j |σi(n) − σj (n)| ≤ S = d X K ℓ=1 (N − (2ℓ − 1)) + R(N − (2K + 1)) = dKN − d X K ℓ=1 (2ℓ − 1) + RN − R(2K + 1) = N 2 − dK2 − KR − KR − R = N 2 − KN − KR − R = N((… view at source ↗
Figure 9
Figure 9. Figure 9: The rotations considered in step (ii) for the special case d = 3. corresponding d-stochastic measure, a straightforward calculation yields Z [0,1]d fmin(x) dϑ ∗ (x) = Z [0,1] fmin(x1, h2(x1), . . . , hd(x1)) dλ(x1) = 1 d as well as Z [0,1]d fmax(x) dϑ ∗ (x) = Z [0,1] fmax(x1, h2(x1), . . . , hd(x1)) dλ(x1) = d − 1 d . In other words: the upper bounds are attainable and the proof is complete. Fig￾ure 10 dep… view at source ↗
Figure 10
Figure 10. Figure 10: Sample (black) of size 1.000 of the 3-stochastic mea￾sure ϑ ∗ attaining the upper bound 1 3 for fmin and 2 3 for fmax (as considered in step (ii) in the proof of Theorem 4.1). The gray/orange/pink points denote the bivariate marginal samples. Theorem 4.2. For the function fP as defined in (4) and every d ≥ 2 we have C d fP = d 2 − 1 3 . Proof. As before we start with sketching the proof structure: (i) Sim… view at source ↗
Figure 11
Figure 11. Figure 11: Aggregation of d = 4 permutations with some overlaps. We partition the grid into d groups of ℓ adjacent rows, i.e., the first ℓ rows form a group, the next ℓ rows another one, and so on. Next we introduce the notion of the ‘level’ of a black block. All blocks in the highest group as well as those in the lowest group have level 1. Blocks located in the second to highest or second to lowest group have level… view at source ↗
Figure 12
Figure 12. Figure 12: Levels of black blocks. Initially, no group is locked. We start with optimizing/shifting the level 1 black blocks and start with those located in the upper group. If there is exactly one of them in each column, then we have nothing to modify und consider this group as locked. Otherwise there is an overlapping or there is one column in which black blocks are at least in two different rows. In case of an ov… view at source ↗
Figure 13
Figure 13. Figure 13: Optimizing the upper level 1 group. We proceed with considering the level 1 black blocks in the lower group. If there is exactly one of them in each column, then we have nothing to modify and consider this group as locked. Otherwise there is again an overlapping or there is one column with black blocks in at least two different rows. In case of an overlapping, we move a black block from a field with multi… view at source ↗
Figure 14
Figure 14. Figure 14: Optimizing the lower level 1 group. Now all level 1 blocks are locked. Note the very important fact that for the current configuration the exact position of level 1 blocks is not relevant for moving blocks of higher level up and down (not entering the locked area) since the sum of the distance to the upper level 1 block and the distance to the lower level 1 block is the same for all possible block positio… view at source ↗
Figure 15
Figure 15. Figure 15: Optimizing the next level. For even dimension d the algorithm stops with locking the two groups of highest level. For odd dimension d the algorithm stops with locking the single group of highest level. Note that if there is only one unlocked group remaining, then this group automatically satisfies that each column contains exactly one black block. In fact, each locked group has exactly one black block per… view at source ↗
Figure 16
Figure 16. Figure 16: Groups with exactly one black block per column. the columns group-wisely does not change the total sum of differences. This allows us to sort the columns in each group such that they form d strictly increasing ‘lines’ of length ℓ (see [PITH_FULL_IMAGE:figures/full_fig_p017_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Groups with reordered columns. (ii) The resulting pattern is obviously induced by the shuffles defined in equation (9) describing equidistant shifting. Altogether we get max ϑ∈Pds([0,1]d) Z [0,1]d fP dϑ = Z [0,1] fP(x1, h2(x1), . . . , hd(x1)) dλ(x1) and it remains to compute the latter integral. A straightforward calculation yields Z [0,1] fP(x1, h2(x1), . . . , hd(x1)) dλ(x1) = fP  0, 1 d , 2 d , . . .… view at source ↗
read the original abstract

This paper was triggered by recent results on the maximal `average distance between consecutive points' of uniformly distributed sequences (u.f.d.s.). Here we address a generalized version of this question, consider pairwise maximal/minimal/total distances in blocks/segments of $ d \geq 2 $ consecutive points of u.f.d.s., and derive sharp upper bounds for all three aggregations. Our main idea of proof consists in, firstly, adding degrees of freedom, secondly, translating the resulting problem to a solvable optimization problem over the compact family of $ d $-stochastic measures, and, thirdly, showing that the obtained bounds are also sharp bounds for the original problem.

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

1 major / 0 minor

Summary. The paper claims to derive sharp upper bounds on the asymptotic average maximal, minimal, and total pairwise distances within blocks of d consecutive points drawn from uniformly distributed sequences, for d ≥ 2. The proof proceeds by relaxing the problem (adding degrees of freedom), solving the resulting optimization over the compact family of d-stochastic measures, and then verifying that the obtained bounds remain sharp when restricted back to blocks arising from actual uniformly distributed sequences.

Significance. If the sharpness verification holds, the results would furnish explicit, sharp constants for these three aggregation functionals on blocks of u.d.s., extending recent work on consecutive-point distances. The relaxation technique to a compact optimization problem over d-stochastic measures supplies a reusable framework that could apply to other extremal questions in uniform distribution theory.

major comments (1)
  1. [Abstract] Abstract (three-step method): the load-bearing claim that the supremum attained over d-stochastic measures equals the limsup of the corresponding quantity over blocks of uniformly distributed sequences is asserted but not substantiated in the provided description; without an explicit construction showing that the optimizing measure arises as a weak-* limit of empirical measures from consecutive blocks of some u.d.s., the numerical value obtained in step (2) may strictly exceed what is attainable in the original setting.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need to ensure the sharpness claim is clearly supported. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (three-step method): the load-bearing claim that the supremum attained over d-stochastic measures equals the limsup of the corresponding quantity over blocks of uniformly distributed sequences is asserted but not substantiated in the provided description; without an explicit construction showing that the optimizing measure arises as a weak-* limit of empirical measures from consecutive blocks of some u.d.s., the numerical value obtained in step (2) may strictly exceed what is attainable in the original setting.

    Authors: We agree the abstract is concise and provides only an outline of the three-step argument. The full manuscript substantiates the equality in the section following the solution of the relaxed optimization problem: an explicit construction is given of a uniformly distributed sequence (via a suitable low-discrepancy sequence whose d-block empirical measures converge in the weak-* topology to the identified optimizer) such that the limsup of the original aggregation functional equals the value attained over d-stochastic measures. This construction ensures the bound obtained in step (2) is attainable in the original setting and is therefore sharp. We are prepared to add a brief sentence to the abstract summarizing the existence of this construction if the referee considers it helpful for clarity. revision: partial

Circularity Check

0 steps flagged

No circularity: relaxation solved independently then sharpness proved separately

full rationale

The derivation relaxes the block-diameter problem by enlarging the feasible set to d-stochastic measures, obtains the optimum over that compact set, and then supplies an independent argument establishing that the same numerical value is attained as a limsup for actual blocks of uniformly distributed sequences. No equation equates the target quantity to a fitted parameter drawn from the same data, no self-citation is invoked as a uniqueness theorem that forces the result, and the sharpness step is not defined circularly in terms of the relaxed optimum itself. The method therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the approach relies on standard compactness of the set of d-stochastic measures and properties of uniform distribution, all treated as background.

pith-pipeline@v0.9.1-grok · 5636 in / 1010 out tokens · 18457 ms · 2026-06-28T20:49:44.172844+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

14 extracted references

  1. [1]

    Bauer Wahrscheinlichkeitstheorie, de Gruyter, Berlin, New York, 2002

    H. Bauer Wahrscheinlichkeitstheorie, de Gruyter, Berlin, New York, 2002

    2002

  2. [2]

    Billingley Convergence of probability measures, Wiley Series in Probability and Statistics: Probability and Statistics John Wiley & Sons Inc., New York, Second edition

    P. Billingley Convergence of probability measures, Wiley Series in Probability and Statistics: Probability and Statistics John Wiley & Sons Inc., New York, Second edition. 1999

    1999

  3. [3]

    Bhattacharya, E

    R. Bhattacharya, E. C. Waymire A Basic Course in Probability Theory, Springer-Verlag, New York, 2007

    2007

  4. [4]

    Drmota, R

    M. Drmota, R. F. Tichy Sequences, Discrepancies and Applications, In: Lecture Notes in Mathematics 1651 , Springer-Verlag, Berlin, 1997

    1997

  5. [5]

    Durante, C

    F. Durante, C. Sempi Copula Theory: An Introduction, Springer, 2015

    2015

  6. [6]

    Fern\'andez S\'anchez, W

    J. Fern\'andez S\'anchez, W. Trutschnig Conditioning based metrics on the space of multivariate copulas and their interrelation with uniform and levelwise convergence and Iterated Function Systems, Journal of Theoretical Probability 28 (2015), 1311--1336

    2015

  7. [7]

    Griessenberger, R.R

    F. Griessenberger, R.R. Junker, W. Trutschnig On a multivariate copula-based dependence measure and its estimation, Electronic Journal of Statistics 16 (2022), 2206--2251

    2022

  8. [8]

    Janssen, J

    P. Janssen, J. Swanepoel, N. Veraverbeke Large sample behavior of the Bernstein copula estimator, Journal of Statistical Planning and Inference 142(5) (2012), 1189--1197

    2012

  9. [9]

    Kallenberg Foundations of modern probability, Springer-Verlag, New York, Berlin, Heidelberg, 1997

    O. Kallenberg Foundations of modern probability, Springer-Verlag, New York, Berlin, Heidelberg, 1997

    1997

  10. [10]

    Klenke Probability Theory - A Comprehensive Course, Springer-Verlag, Berlin, Heidelberg, 2007

    A. Klenke Probability Theory - A Comprehensive Course, Springer-Verlag, Berlin, Heidelberg, 2007

    2007

  11. [11]

    Kuipers, H

    L. Kuipers, H. Niederreiter Uniform Distribution of Sequences, Wiley, New York, 1974; reprint, Dover Publications, Mineola, NY, 2006

    1974

  12. [12]

    Kunze, D

    H. Kunze, D. La Torre, F. Mendivil, E. R. Vrscay Fractal-Based Methods in Analysis, Springer Science & Business Media, 2011

    2011

  13. [13]

    Pillichshammer, S

    F. Pillichshammer, S. Steinerberger Average distance between consecutive points of uniformly distributed sequences, Unif. Distrib. Theory 4(1) (2009), 51--67

    2009

  14. [14]

    van der Vaart, J.A

    A. van der Vaart, J.A. Wellner Weak Convergence and Empirical Processes, Springer Series in Statistics, 1996

    1996