the center of distances for some multigeometric series

Artur Bartoszewicz; Franciszek Prus-Wi\'sniowski; Micha{\l} Banakiewicz

arxiv: 1907.03800 · v1 · pith:S6TOGUS4new · submitted 2019-07-08 · 🧮 math.CA

the center of distances for some multigeometric series

Micha{\l} Banakiewicz , Artur Bartoszewicz , Franciszek Prus-Wi\'sniowski This is my paper

Pith reviewed 2026-05-25 00:37 UTC · model grok-4.3

classification 🧮 math.CA

keywords center of distancesmultigeometric seriesachievement setssubsumsconvergent seriesdistance sets

0 comments

The pith

For some multigeometric series the center of distances of the set of subsums is the union of centers from its qualifying initial partial sums.

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

The paper studies the center of distances for sets, with emphasis on achievement sets from series. It finds a sufficient condition under which this center for fast convergent series contains only zero and the series terms. For particular multigeometric series it establishes that the center is the union of the centers of distances of the sets of n-initial subsums that satisfy special requirements.

Core claim

A complete description shows that the center of distances C(E) is the union of all centers of distances of C(F_n) where F_n is the set of all n-initial subsums of the multigeometric series satisfying some special requirements.

What carries the argument

Union of the centers of distances C(F_n) over qualifying initial subsum sets F_n of the multigeometric series.

If this is right

The center of distances can be determined from the centers of its finite initial subsum sets.
A new sufficient condition is given for the center to consist only of 0 and the series terms in fast convergent cases.
Several open questions are raised about the general structure of such centers.
Basic properties of the center of distances are investigated for achievement sets.

Where Pith is reading between the lines

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

This decomposition suggests that the property may extend to other classes of series with similar convergence behavior.
Verification for concrete examples of multigeometric series could confirm the union structure in practice.
Connections to other set-theoretic properties of subsums might be explored using this reduction.

Load-bearing premise

The multigeometric series must satisfy special requirements that make the union of the initial centers describe the full center.

What would settle it

Take a specific multigeometric series meeting the requirements, compute its center of distances directly, and check whether it matches the union of the centers computed from its n-initial subsums.

read the original abstract

Basic properties of the center of distances of a set are investigated. Computation of the center for achievement sets is particularly aimed at. A new sufficient condition for the center of distances of the set of subsums of a fast convergent series to consist of only 0 and the terms of the series is found. A complete description of the center of distances for some particular type of multigeometric series is provided. In particular, the center of distances $C(E)$ is the union of all centers of distances of $C(F_n)$ where $F_n$ is the set of all $n$-initial subsums of the discussed multigeometric series satisfying some special requirements. Several open questions are raised.

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.

Desk Editor's Note private letter to a colleague

Paper gives an explicit union-of-centers description for the distance center of one narrow class of multigeometric series, plus a sufficient condition for fast convergent series.

read the letter

This paper pins down the center of distances C(E) for a specific family of multigeometric series by writing it as the union of C(F_n) over the n-initial subsums that meet the stated special requirements. It also gives a new sufficient condition under which the center for the achievement set of a fast convergent series collapses to just {0} together with the series terms themselves. Both pieces are concrete and scoped to the constructions they consider. The opening discussion of basic properties of centers helps frame the later results, and the open questions at the end are a reasonable way to close. The union description looks like it follows directly from the definitions once the requirements are fixed, so the central claim should be checkable by direct verification. The main limitation is the narrow target. Everything is restricted to this one type of series and the special requirements that make the union work; there is no exploration of how often those requirements hold more generally or how the result compares with other series classes. No downstream applications or links to wider questions in the literature are developed, so the payoff stays inside the subfield. Readers who already work on achievement sets or distance centers in real analysis will find a usable explicit case. The work is grounded enough in the existing notions to deserve referee time; the claims are specific enough that experts can test the proofs and the requirements without needing major new machinery.

Referee Report

0 major / 3 minor

Summary. The manuscript investigates basic properties of the center of distances C(E) for a set E. It focuses on achievement sets arising as sets of subsums of series. A new sufficient condition is given under which C(E) for the achievement set of a fast convergent series consists only of 0 and the series terms. For a specific class of multigeometric series, a complete description is claimed: C(E) equals the union over n of C(F_n), where each F_n is the set of n-initial subsums satisfying explicitly stated special requirements. Several open questions are raised.

Significance. If the stated description and condition hold, the results supply concrete characterizations of centers of distances for achievement sets of multigeometric series. This contributes to the literature on additive structure and distance sets in the context of convergent series, potentially aiding computations in related problems in geometric measure theory.

minor comments (3)

The abstract refers to 'some special requirements' for the subsums F_n; the main text should state these requirements explicitly at the outset of the relevant section and verify that they are satisfied by the series under consideration.
The paper should include at least one fully worked numerical example computing both sides of the claimed equality C(E) = ∪ C(F_n) to illustrate the union description.
Notation for the multigeometric series and the initial subsums F_n should be introduced with a clear definition before the statement of the main theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript on centers of distances for achievement sets of multigeometric series and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states basic properties of centers of distances, gives a new sufficient condition for achievement sets of fast convergent series, and claims a complete description of C(E) for a specific class of multigeometric series as the union of C(F_n) over qualifying n-initial subsums. No equations, definitions, or claims in the provided abstract reduce any prediction or central result to a fitted input, self-definition, or self-citation chain; the description is scoped to explicitly constructed series meeting stated requirements, with both inclusions presumably proved directly from the series structure. The work is self-contained against external benchmarks of achievement-set theory and raises open questions, indicating independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted from the provided text.

pith-pipeline@v0.9.0 · 5653 in / 988 out tokens · 21262 ms · 2026-05-25T00:37:27.028244+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

A complete description of the center of distances for some particular type of multigeometric series is provided. In particular, the center of distances C(E) is the union of all centers of distances of C(F_n) where F_n is the set of all n-initial subsums...
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 4.1... C(E) = ⋃_{k=0}^∞ q^k C(F_m)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

[1]

Banakh, A

T. Banakh, A. Bartoszewicz, M. Filipczak, E. Szymonik, Topological and measure properties of some self-similar sets , Topol. Methods Nonliner Analysis 46(2015) 1013-1028

work page 2015
[2]

Banakiewicz, F

M. Banakiewicz, F. Prus-Wi´ sniowski, M-Cantorvals of Ferens type , Math. Slovaka 67(4)(2017) 1-12

work page 2017
[3]

G/suppress l¸ ab, M

A.Bartoszewicz, Sz. G/suppress l¸ ab, M. Filipczak, F. Prus-Wi´ sniowski, J. Swaczyna, On generating reg- ular Cantorvals connected with geometric Cantor sets , Chaos, Solitons and Fractals 114(2018) 468-473

work page 2018
[4]

Bartoszewicz, Sz

A. Bartoszewicz, Sz. G/suppress l¸ ab, J. Marchwicki,Recovering purely atomic ﬁnite measure from its range, J. Math. Anal. Appl. 467(2)(2018) 825-841

work page 2018
[5]

Filipczak, F

A.Bartoszewicz, M. Filipczak, F. Prus-Wi´ sniowski, Topological and algebraic aspects of sub- sums of series , in: Traditional and Present-day Topipcs in Real Analysis , Faculty of Mathe- matics and Computer Science, University of /suppress L´ od´ z, /suppress L´ od´z, 2013, pp. 345-366

work page 2013
[6]

Filipczak, F

A.Bartoszewicz, M. Filipczak, F. Prus-Wi´ sniowski, Semi-fast convergent series and k-sums of central Cantor sets , submitted

work page
[7]

Bartoszewicz, M

A. Bartoszewicz, M. Filipczak, E. Szymonik, Multigeometric sequences and Cantorvals , Centr. Eur. J. Math. 12(7)(2014) 1000-1007

work page 2014
[8]

Bielas, Sz

W. Bielas, Sz. Plewik, M. W alczy´ nska, On the center of distances , European J. Math. 4(2)(2018) 687-698

work page 2018
[9]

Ferens, On the range of purely atomic measures , Studia Math

C. Ferens, On the range of purely atomic measures , Studia Math. 77(1984) 261-263

work page 1984
[10]

Guthrie, J.E

J.A. Guthrie, J.E. Nymann, The topological structure of the set of subsums of an inﬁnite series, Colloq. Math. 55(1988) 323-327

work page 1988
[11]

Halmos, Permutations of sequences and the Schr¨ oder-Bernstein the orem, Proc

P.R. Halmos, Permutations of sequences and the Schr¨ oder-Bernstein the orem, Proc. Amer. Math. Soc. 19(1968) 509-510

work page 1968
[12]

Jones, Achievement sets of sequences , Amer

R. Jones, Achievement sets of sequences , Amer. Math. Monthly 122(9)(2015) 862-870

work page 2015
[13]

Kakeya, On the partial sums of an inﬁnite series , Tˆ ohoku Sci

S. Kakeya, On the partial sums of an inﬁnite series , Tˆ ohoku Sci. Rep. 3, no. 4 (1914), 159-164

work page 1914
[14]

Kakeya, On the set of partial sums of an inﬁnite series , Proc

S. Kakeya, On the set of partial sums of an inﬁnite series , Proc. Tokyo Math.-Phys. Soc. 2nd ser. 7(1914), 250-251

work page 1914
[15]

P.Mendes, F.Oliveira, On the topological structure of the arithmetic sum of two Can tor sets , Nonlinearity 7(1994) 329-343

work page 1994
[16]

Subsum Sets: Intervals, Cantor Sets, and Cantorvals

Z. Nitecki, The subsum set of a null sequence , arXiv: 1106.3779v1

work page internal anchor Pith review Pith/arXiv arXiv
[17]

Nitecki, Cantorvals and Subsum Sets of Null Sequences , Amer

Z. Nitecki, Cantorvals and Subsum Sets of Null Sequences , Amer. Math. Monthly 122(9)(2015) 862-870

work page 2015
[18]

von Neumann, Characterizirung des Spectrums eines Integraloperators , Hermann, Paris 1935

J. von Neumann, Characterizirung des Spectrums eines Integraloperators , Hermann, Paris 1935

work page 1935
[19]

Nymann, R.A

J.E. Nymann, R.A. S´ aenz, On the paper of Guthrie and Nymann on subsums of an inﬁnite series, Colloq. Math. 83(2000) 1-4

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F. Prus-Wi´ sniowski,Beyond the sets of subsums , preprints of the Faculty of Matematics and Informatics, /suppress L´ od´ z University 2013; www.math.uni.lodz.pl/preprints,all.html 12 M. BANAKIEWICZ, A. BARTOSZEWICZ AND F. PRUS-WI ´SNIOWSKI * Studium of Mathematics West Pomeranian University of Technology Al. Piast ´ow 48 PL-70-311 Szczecin Poland E-mail...

work page 2013

[1] [1]

Banakh, A

T. Banakh, A. Bartoszewicz, M. Filipczak, E. Szymonik, Topological and measure properties of some self-similar sets , Topol. Methods Nonliner Analysis 46(2015) 1013-1028

work page 2015

[2] [2]

Banakiewicz, F

M. Banakiewicz, F. Prus-Wi´ sniowski, M-Cantorvals of Ferens type , Math. Slovaka 67(4)(2017) 1-12

work page 2017

[3] [3]

G/suppress l¸ ab, M

A.Bartoszewicz, Sz. G/suppress l¸ ab, M. Filipczak, F. Prus-Wi´ sniowski, J. Swaczyna, On generating reg- ular Cantorvals connected with geometric Cantor sets , Chaos, Solitons and Fractals 114(2018) 468-473

work page 2018

[4] [4]

Bartoszewicz, Sz

A. Bartoszewicz, Sz. G/suppress l¸ ab, J. Marchwicki,Recovering purely atomic ﬁnite measure from its range, J. Math. Anal. Appl. 467(2)(2018) 825-841

work page 2018

[5] [5]

Filipczak, F

A.Bartoszewicz, M. Filipczak, F. Prus-Wi´ sniowski, Topological and algebraic aspects of sub- sums of series , in: Traditional and Present-day Topipcs in Real Analysis , Faculty of Mathe- matics and Computer Science, University of /suppress L´ od´ z, /suppress L´ od´z, 2013, pp. 345-366

work page 2013

[6] [6]

Filipczak, F

A.Bartoszewicz, M. Filipczak, F. Prus-Wi´ sniowski, Semi-fast convergent series and k-sums of central Cantor sets , submitted

work page

[7] [7]

Bartoszewicz, M

A. Bartoszewicz, M. Filipczak, E. Szymonik, Multigeometric sequences and Cantorvals , Centr. Eur. J. Math. 12(7)(2014) 1000-1007

work page 2014

[8] [8]

Bielas, Sz

W. Bielas, Sz. Plewik, M. W alczy´ nska, On the center of distances , European J. Math. 4(2)(2018) 687-698

work page 2018

[9] [9]

Ferens, On the range of purely atomic measures , Studia Math

C. Ferens, On the range of purely atomic measures , Studia Math. 77(1984) 261-263

work page 1984

[10] [10]

Guthrie, J.E

J.A. Guthrie, J.E. Nymann, The topological structure of the set of subsums of an inﬁnite series, Colloq. Math. 55(1988) 323-327

work page 1988

[11] [11]

Halmos, Permutations of sequences and the Schr¨ oder-Bernstein the orem, Proc

P.R. Halmos, Permutations of sequences and the Schr¨ oder-Bernstein the orem, Proc. Amer. Math. Soc. 19(1968) 509-510

work page 1968

[12] [12]

Jones, Achievement sets of sequences , Amer

R. Jones, Achievement sets of sequences , Amer. Math. Monthly 122(9)(2015) 862-870

work page 2015

[13] [13]

Kakeya, On the partial sums of an inﬁnite series , Tˆ ohoku Sci

S. Kakeya, On the partial sums of an inﬁnite series , Tˆ ohoku Sci. Rep. 3, no. 4 (1914), 159-164

work page 1914

[14] [14]

Kakeya, On the set of partial sums of an inﬁnite series , Proc

S. Kakeya, On the set of partial sums of an inﬁnite series , Proc. Tokyo Math.-Phys. Soc. 2nd ser. 7(1914), 250-251

work page 1914

[15] [15]

P.Mendes, F.Oliveira, On the topological structure of the arithmetic sum of two Can tor sets , Nonlinearity 7(1994) 329-343

work page 1994

[16] [16]

Subsum Sets: Intervals, Cantor Sets, and Cantorvals

Z. Nitecki, The subsum set of a null sequence , arXiv: 1106.3779v1

work page internal anchor Pith review Pith/arXiv arXiv

[17] [17]

Nitecki, Cantorvals and Subsum Sets of Null Sequences , Amer

Z. Nitecki, Cantorvals and Subsum Sets of Null Sequences , Amer. Math. Monthly 122(9)(2015) 862-870

work page 2015

[18] [18]

von Neumann, Characterizirung des Spectrums eines Integraloperators , Hermann, Paris 1935

J. von Neumann, Characterizirung des Spectrums eines Integraloperators , Hermann, Paris 1935

work page 1935

[19] [19]

Nymann, R.A

J.E. Nymann, R.A. S´ aenz, On the paper of Guthrie and Nymann on subsums of an inﬁnite series, Colloq. Math. 83(2000) 1-4

work page 2000

[20] [20]

F. Prus-Wi´ sniowski,Beyond the sets of subsums , preprints of the Faculty of Matematics and Informatics, /suppress L´ od´ z University 2013; www.math.uni.lodz.pl/preprints,all.html 12 M. BANAKIEWICZ, A. BARTOSZEWICZ AND F. PRUS-WI ´SNIOWSKI * Studium of Mathematics West Pomeranian University of Technology Al. Piast ´ow 48 PL-70-311 Szczecin Poland E-mail...

work page 2013