On the number of finite additive 2-bases

Konrad Zyhalko; Stefan Weltge

arxiv: 2605.19449 · v2 · pith:CRBNKEAHnew · submitted 2026-05-19 · 🧮 math.CO · cs.DM

On the number of finite additive 2-bases

Stefan Weltge , Konrad Zyhalko This is my paper

Pith reviewed 2026-05-22 10:04 UTC · model grok-4.3

classification 🧮 math.CO cs.DM

keywords additive 2-basesfinite setssumsetsexponential growthprobabilistic methodadditive combinatorics

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The pith

The number of finite additive 2-bases grows exponentially.

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

This paper establishes that the number of finite additive 2-bases grows exponentially. It does so with a direct short proof that relies only on elementary probabilistic arguments. The result was previously obtained through complex analytic techniques in the study of numerical sets. A reader would care because the simpler argument makes the combinatorial abundance of these sets more transparent and easier to verify.

Core claim

The number of finite additive 2-bases is known to grow exponentially, and we establish this fact with a direct short proof using elementary probabilistic arguments.

What carries the argument

Elementary probabilistic counting argument that bounds the proportion of integer sets whose pairwise sums cover a consecutive interval.

If this is right

There exist at least exponentially many distinct finite sets A whose sumset A+A covers every integer in some interval of length proportional to |A|.
The exponential growth holds already for bases of any fixed span once the size parameter is large.
The same probabilistic technique supplies a lower bound without invoking generating functions or residue-class analysis.

Where Pith is reading between the lines

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

The method may extend to counting restricted additive bases with additional sum-free or Sidon-type conditions.
A matching upper bound on the growth rate would require different tools and remains open from this argument alone.

Load-bearing premise

The probabilistic estimates correctly bound the count of valid 2-bases without hidden dependencies or overcounting that would invalidate the exponential lower bound.

What would settle it

An explicit enumeration or tighter counting argument for large intervals showing only polynomially many such 2-bases would disprove the exponential growth.

read the original abstract

The number of finite additive 2-bases is known to grow exponentially. While this fact has been established by Marzuola and Miller (2010) using complex analytic techniques embedded in the study of numerical sets, we provide a direct, short proof using elementary probabilistic arguments.

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

This paper gives a simpler probabilistic proof of the known exponential growth in the number of finite additive 2-bases.

read the letter

The key takeaway is that this work supplies an alternative, elementary proof that the number of finite additive 2-bases grows exponentially with the size of the underlying interval. This fact was already established by Marzuola and Miller in 2010 using more involved analytic techniques, so the main contribution here is the new proof method rather than a new result. The paper does well in keeping the argument direct and accessible. By using basic probabilistic ideas to construct or count many such bases, it sidesteps the complex analysis and makes the exponential growth easier to see for those not steeped in that machinery. This kind of simplification can be genuinely useful in the subfield. The soft spots are around the probabilistic estimates themselves. The stress-test note points out a possible issue with positive correlations in the coverage events for the sums. In random subset models, it's common for the events that particular sums are covered to be dependent, and a plain first-moment calculation can overestimate the probability of success. Without something like the Lovász local lemma or a deletion step to handle the dependencies, the lower bound on the number might not reach exponential. The abstract promises only elementary methods, which often means union bound or direct expectation, so this is the part that needs close checking in the full text. If the paper has a clean way around it, then the proof holds up; otherwise, it's a real but fixable concern. The work is not circular and stands on its own probabilistic derivation. This paper is for researchers in additive combinatorics who want to understand or teach this counting result without heavy tools. A reader interested in new phenomena or broader applications won't find them here, but it could serve as a reference for the probabilistic approach. It deserves peer review because the core claim is clear and the method is promising, even if the details of the estimates need verification. I recommend sending it to referees.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that the number of finite additive 2-bases grows exponentially and supplies a short, direct proof of this fact via elementary probabilistic arguments on random subsets, in contrast to the complex-analytic methods of Marzuola and Miller (2010).

Significance. If the probabilistic lower bound is rigorously established, the result offers a simpler, self-contained argument for exponential growth in the count of finite additive 2-bases, which may facilitate extensions to related problems in additive combinatorics and make the fact more accessible without analytic machinery.

major comments (1)

[Section 2] Section 2 (probabilistic construction): the first-moment calculation on the coverage indicators for sums in [1,2N] does not address possible positive correlations arising when the same random elements contribute to multiple sums; without an explicit second-moment bound, variance estimate, or deletion step, the claimed success probability may be overestimated and the resulting exponential lower bound on the number of 2-bases could fail to hold.

minor comments (2)

[Abstract] The abstract and introduction should explicitly state the precise definition of 'finite additive 2-basis' used (e.g., whether A+A covers [1,2 max A] or all of Z up to a fixed length).
[Section 2] Notation for the random model (density parameter, interval size N) should be introduced once and used consistently throughout the estimates.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting a potential issue with dependencies in the probabilistic argument of Section 2. We address the comment directly below.

read point-by-point responses

Referee: [Section 2] Section 2 (probabilistic construction): the first-moment calculation on the coverage indicators for sums in [1,2N] does not address possible positive correlations arising when the same random elements contribute to multiple sums; without an explicit second-moment bound, variance estimate, or deletion step, the claimed success probability may be overestimated and the resulting exponential lower bound on the number of 2-bases could fail to hold.

Authors: We appreciate this observation on possible correlations. Our argument proceeds by letting X be the number of uncovered sums in [1,2N] for a random subset A, where each element of [N] is included independently with a suitable probability p. We compute E[X] explicitly via linearity and show that E[X] is bounded by a constant less than 1 (or sufficiently small relative to the total number of subsets). The probability that A fails to be a 2-basis is then at most E[X] by Markov's inequality (equivalently, the union bound on the events that individual sums are uncovered). This upper bound on the failure probability holds unconditionally and does not rely on independence or absence of correlation among the indicators; positive correlations only affect the variance of X but leave the first-moment bound intact. Consequently the success probability is at least 1 - E[X] > 0, which immediately yields the exponential lower bound on the total number of finite additive 2-bases. No second-moment estimate or deletion step is required for this conclusion. We will add a clarifying sentence in the revised version of Section 2 to state explicitly that the bound follows from Markov's inequality and is therefore robust to dependencies. revision: partial

Circularity Check

0 steps flagged

Independent elementary probabilistic proof with no reduction to inputs

full rationale

The paper cites Marzuola and Miller (2010) solely to contrast their complex analytic approach with the new direct probabilistic argument; the exponential lower bound is derived afresh via sampling random subsets and bounding the probability that they form additive 2-bases. No equations reduce a claimed count to a fitted parameter, no self-citation is load-bearing for the main result, and the derivation stands on its own elementary estimates without importing uniqueness theorems or ansatzes from the authors' prior work. The central claim therefore remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the applicability of the probabilistic method to counting problems in additive combinatorics, which is a standard domain tool rather than a new axiom invented here.

axioms (1)

domain assumption Elementary probabilistic arguments suffice to establish a lower bound on the number of finite additive 2-bases.
The abstract invokes probabilistic arguments as the core of the short proof.

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Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

Güntürk and M

C. Güntürk and M. Nathanson. A new upper bound for finite additive bases.Acta Arithmetica, 124:235–255, 2006

work page 2006
[2]

Hämmerer and G

N. Hämmerer and G. Hofmeister. Zu einer Vermutung von Rohrbach.Journal für die reine und angewandte Mathematik, 286/287:239–247, 1976

work page 1976
[3]

Counting numerical sets with no small atoms

Jeremy Marzuola and Andy Miller. Counting numerical sets with no small atoms. Journal of Combinatorial Theory, Series A, 117(6):650–667, 2010

work page 2010
[4]

L. Moser. On the representation of 1, 2, . . . , n by sums.Acta Arithmetica, 6:11–13, 1960

work page 1960
[5]

A. Mrose. Untere Schranken für die Reichweiten von Extremalbasen fester Ord- nung.Abhandlungen aus dem Mathematischen Seminar der Universität Ham- burg, 48:118–124, 1979

work page 1979
[6]

The On-Line Encyclopedia of Integer Sequences.http: //oeis.org, 2023

OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences.http: //oeis.org, 2023

work page 2023
[7]

Rohrbach

H. Rohrbach. Ein Beitrag zur additiven Zahlentheorie.Mathematische Zeitschrift, pages 1–30, 1937

work page 1937
[8]

G. Yu. Upper bounds for finite additive 2-bases.Journal of the American Mathe- matical Society, 137:11–18, 2009. 4

work page 2009

[1] [1]

Güntürk and M

C. Güntürk and M. Nathanson. A new upper bound for finite additive bases.Acta Arithmetica, 124:235–255, 2006

work page 2006

[2] [2]

Hämmerer and G

N. Hämmerer and G. Hofmeister. Zu einer Vermutung von Rohrbach.Journal für die reine und angewandte Mathematik, 286/287:239–247, 1976

work page 1976

[3] [3]

Counting numerical sets with no small atoms

Jeremy Marzuola and Andy Miller. Counting numerical sets with no small atoms. Journal of Combinatorial Theory, Series A, 117(6):650–667, 2010

work page 2010

[4] [4]

L. Moser. On the representation of 1, 2, . . . , n by sums.Acta Arithmetica, 6:11–13, 1960

work page 1960

[5] [5]

A. Mrose. Untere Schranken für die Reichweiten von Extremalbasen fester Ord- nung.Abhandlungen aus dem Mathematischen Seminar der Universität Ham- burg, 48:118–124, 1979

work page 1979

[6] [6]

The On-Line Encyclopedia of Integer Sequences.http: //oeis.org, 2023

OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences.http: //oeis.org, 2023

work page 2023

[7] [7]

Rohrbach

H. Rohrbach. Ein Beitrag zur additiven Zahlentheorie.Mathematische Zeitschrift, pages 1–30, 1937

work page 1937

[8] [8]

G. Yu. Upper bounds for finite additive 2-bases.Journal of the American Mathe- matical Society, 137:11–18, 2009. 4

work page 2009