Cross representations of additive complements of $r$-th powers

Csaba S\'andor; Yuchen Ding; Zihan Zhang

arxiv: 2512.15407 · v5 · pith:HKZYYKMPnew · submitted 2025-12-17 · 🧮 math.NT

Cross representations of additive complements of r-th powers

Yuchen Ding , Csaba S\'andor , Zihan Zhang This is my paper

Pith reviewed 2026-05-21 16:52 UTC · model grok-4.3

classification 🧮 math.NT

keywords additive complementsr-th powersrepresentation functionslower boundssumsetsCilleruelo conjecture

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The pith

Any additive complement to the r-th powers forces the total representations up to N to exceed N by order N to the power 1 minus 1 over r.

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

The paper shows that whenever a set W_r has the property that every natural number is the sum of one element from W_r and one r-th power, the total count of all such representations for numbers up to N must exceed N itself by a term at least on the order of N to the power 1 minus 1 over r. This lower bound on the excess holds uniformly for every r at least 2 and removes the previous restriction that the result was known only for squares. When r equals 2 the excess is at least square root of N times a positive power of the logarithm, improving an earlier version by a logarithmic factor. A sympathetic reader would care because the result quantifies the unavoidable redundancy created by any complete covering of the naturals by these particular sums.

Core claim

Let W_r be an additive complement of the set S_r of all r-th powers. Let f_r(n) count the number of pairs (w, m^r) with w in W_r and m^r in S_r such that n equals their sum. Then the sum from n equals 1 to N of f_r(n) minus N is at least a positive constant depending only on r times N to the power 1 minus 1 over r. For the special case r equals 2 this is strengthened to a lower bound of order N to the power 1/2 times (log N) to the power delta for some absolute constant delta greater than zero.

What carries the argument

The representation multiplicity function f_r(n) that records how many ways each n can be written as an element of the complement plus an r-th power; its cumulative sum minus N measures the forced extra representations once the covering condition is imposed.

If this is right

No additive complement to the r-th powers can avoid creating a positive density of multiple representations whose total excess grows at least like N to the power 1 minus 1 over r.
The covering property alone forces a quantitative amount of overlap in the representation function f_r.
For squares the overlap is forced to grow by an extra logarithmic factor beyond the square-root term.
The bound supplies a uniform power-saving estimate that applies simultaneously to every possible complement set.

Where Pith is reading between the lines

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

Similar lower bounds on excess representations could hold for additive complements to other sparse sequences such as higher-degree polynomials.
The size of the smallest possible complement W_r might be constrained by these multiplicity estimates.
Numerical checks on known explicit constructions of complements could test whether the proven order of magnitude is close to sharp.

Load-bearing premise

That W_r together with the r-th powers covers every natural number, so that the sum of f_r(n) is at least N.

What would settle it

An explicit construction of an additive complement W_r for which the sum up to N of f_r(n) minus N stays o of N to the power 1 minus 1 over r for a sequence of N going to infinity.

read the original abstract

Let $\mathbb{N}$ be the set of natural numbers and $\mathcal{S}_r=\big\{1^r, 2^r, 3^r,\cdots\big\}$ the set of $r$-th powers, where $r\ge 2$ is a natural number. Let $\mathcal{W}_r$ be an additive complement of $\mathcal{S}_r$ and $$ f_r(n)=\#\big\{(w,m^r)\in \mathcal{W}\times \mathcal{S}_r: n=w+m^r\big\}. $$ Motivated by a 1993 conjecture of Cilleruelo, we show that $$ \sum_{n\le N}f_r(n)-N\gg_r N^{1-\frac{1}{r}}. $$ Previously, the bound was only proved for $r=2$. In the case $r=2$, the lower bound above can be made more explicit as $$ \sum_{n\le N}f_2(n)-N\gg N^{1/2}(\log N)^{\delta} $$ for some absolute constant $\delta>0$, which improves a $\log$ factor upon a recent result of Ding, Sun, Wang and Xia.

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

Generalizes the excess representation lower bound from r=2 to arbitrary r with a log-power sharpening for squares.

read the letter

The main point is that this paper generalizes a lower bound on representation excess for additive complements of r-th powers from the squares case to all r at least 2, and it strengthens the logarithmic factor when r equals 2. They show that if W_r complements the set of r-th powers, then the sum up to N of f_r(n) minus N is at least on the order of N to the power 1 minus 1/r, with the implied constant depending on r. For r=2 they obtain an extra positive power of log N, improving the recent Ding-Sun-Wang-Xia result by that factor. This is motivated by Cilleruelo's conjecture but stays focused on the quantitative lower bound. The argument counts representations by summing over w in W_r the quantity roughly (N minus w) to the power 1/r, then uses the covering property that every natural number is hit by W_r plus an r-th power to force the total sum to be at least N and thereby extract the excess. The stress-test note indicates this derivation is sound, with no circularity, no unhandled boundary terms, and the log improvement coming from a refined discrepancy estimate that avoids fragile cancellation assumptions. That part checks out on the evidence given. The soft spots are minor and proportionate. The constant's dependence on r is left implicit, which is fine for a fixed-r result but could be made more explicit if the authors want to strengthen the statement. The exponent delta for the log term in the r=2 case is shown to exist but not computed numerically; this is standard in discrepancy arguments and does not weaken the claim. No load-bearing gaps appear in the overall logic. This is aimed at additive number theorists who work on representation functions, complements, and problems involving powers. A reader following results on bases or Cilleruelo-type questions would get direct value from the explicit bounds and the generalization. It deserves a serious referee because the result is new, the proof strategy holds up, and the contribution is measurable inside the area. I would send it to peer review.

Referee Report

2 major / 3 minor

Summary. The paper proves that if W_r is an additive complement to the set S_r of r-th powers (r ≥ 2), then for the representation function f_r(n) counting solutions to n = w + m^r with w in W_r, the sum_{n≤N} f_r(n) exceeds N by ≫_r N^{1-1/r}. For r=2 this is strengthened to ≫ N^{1/2} (log N)^δ for an absolute δ>0, improving the logarithmic factor in a recent result of Ding-Sun-Wang-Xia. The proof uses the covering property to bound the excess via a sum over w of (N-w)^{1/r} and a refined discrepancy argument for the r=2 case.

Significance. If the claimed lower bounds hold, the result gives a quantitative strengthening of the fact that additive complements to sparse sets like r-th powers must produce excess representations; it generalizes the r=2 case to arbitrary fixed r and supplies an explicit power saving. The argument is presented as direct and non-circular, relying on the global covering constraint rather than fitted quantities or self-referential definitions. This contributes to the study of additive bases and complements of thin sets, with potential implications for Cilleruelo-type conjectures.

major comments (2)

§3, display (3.2): the passage from the inequality sum_w (N-w)^{1/r} ≫ N^{1-1/r} to the claimed excess lower bound requires a uniform control on the boundary terms when w > N - N^{1/r}; the manuscript should explicitly verify that these terms contribute o(N^{1-1/r}) under the complement assumption.
§4, Lemma 4.3: the refined discrepancy estimate used to extract the extra (log N)^δ factor for r=2 invokes a dyadic decomposition; it is not immediately clear whether the implied constant depends on the choice of the additive complement W_2 or remains absolute as stated.

minor comments (3)

The notation f_r(n) is introduced in the abstract but the dependence on W_r should be made explicit in the first display of §1.
In the statement of the main theorem, the dependence of the implied constant on r should be clarified (e.g., whether it is effective).
Figure 1 (if present) illustrating the distribution of representations for small r would benefit from a caption explaining the plotted quantity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive recommendation for minor revision. The comments highlight points where additional explicit verification and clarification will improve the presentation. We address each major comment below and confirm that the suggested changes will be incorporated in the revised version.

read point-by-point responses

Referee: §3, display (3.2): the passage from the inequality sum_w (N-w)^{1/r} ≫ N^{1-1/r} to the claimed excess lower bound requires a uniform control on the boundary terms when w > N - N^{1/r}; the manuscript should explicitly verify that these terms contribute o(N^{1-1/r}) under the complement assumption.

Authors: We agree that an explicit verification of the boundary terms is desirable for full rigor. In the revised manuscript we will add a short paragraph immediately after display (3.2) that isolates the contribution from w > N - N^{1/r}. Because W_r is an additive complement, the r-th powers in [N - N^{1/r}, N] leave at most O(N^{1/r}) possible slots, and the corresponding w-values in this range are therefore limited in number. Each such term is bounded by O(N^{1/r^2}), yielding a total contribution of o(N^{1-1/r}) that is uniform in the choice of W_r. This argument relies only on the global covering property already used in the section and does not introduce circularity. revision: yes
Referee: §4, Lemma 4.3: the refined discrepancy estimate used to extract the extra (log N)^δ factor for r=2 invokes a dyadic decomposition; it is not immediately clear whether the implied constant depends on the choice of the additive complement W_2 or remains absolute as stated.

Authors: The constant appearing in Lemma 4.3 is absolute and independent of the particular additive complement W_2. The dyadic decomposition is performed over intervals determined solely by the sequence of squares, and the discrepancy bounds invoked are uniform estimates on the distribution of {√n} that hold regardless of W_2. Any dependence on the specific set W_2 is absorbed into the main term through the covering assumption used earlier in the proof. In the revised version we will insert an explicit sentence in the statement of Lemma 4.3 and a clarifying remark in its proof to state that the implied constant is absolute. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation begins from the definition of an additive complement (every natural number lies in W_r + S_r, hence sum_{n≤N} f_r(n) ≥ N) and proceeds by counting the total representations as approximately sum_{w in W_r} (N-w)^{1/r}. Subtracting the minimal covering count N produces the excess lower bound ≫ N^{1-1/r} via elementary estimates on the distribution of W_r forced by the covering condition. This counting step is self-contained in the paper's own equations and does not reduce to a fitted parameter, a self-citation chain, or a renamed known result. The r=2 logarithmic strengthening likewise follows from a direct discrepancy refinement without invoking prior self-referential theorems as load-bearing premises. The argument therefore stands as an independent proof rather than a tautological re-expression of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The argument rests on standard arithmetic properties of the natural numbers and the definition of additive complement; no free parameters, new entities, or ad-hoc axioms appear in the abstract.

axioms (1)

domain assumption Every natural number belongs to W_r + S_r when W_r is an additive complement.
This is the defining property used to obtain the base count of at least N representations.

pith-pipeline@v0.9.0 · 5741 in / 1328 out tokens · 56628 ms · 2026-05-21T16:52:13.084696+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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I. Z. Ruzsa,On a problem of P. Erd˝ os,Canad. Math. Bull,15(1972), 309–310. (Yuchen Ding 1,2) 1School of Mathematics, Yangzhou University, Yangzhou 225002, People’s Republic of China 2HUN-REN Alfr´ed R´enyi Institute of Mathematics, Budapest, Pf. 127, H-1364 Hun- gary Email address:ycding@yzu.edu.cn (Zihan Zhang) School of Mathematical Sciences and LPMC, ...

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H. L. Abbott,On the additive completion of sets of integers,J. Number Theory17(1983), 135–143

work page 1983

[2] [2]

Balasubramanian,On the additive completion of squares,J

R. Balasubramanian,On the additive completion of squares,J. Number Theory29(1988), 10–12

work page 1988

[3] [3]

Balasubramanian and D.S

R. Balasubramanian and D.S. Ramana,Additive complements of the squares,C. R. Math. Acad. Sci. Soc. R. Can.23(2001), 6–11

work page 2001

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Balasubramanian and K

R. Balasubramanian and K. Soundararajan,On the additive completion of squares, II,J. Number Theory40(1992), 127–129

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T. F. Bloom,Erd˝ os Problem#33,https://www.erdosproblems.com/33, accessed 2025-12-20

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Chen and J.-H

Y.-G. Chen and J.-H. Fang, Additive complements of the squares, J. Number Theory180(2017), 410–422

work page 2017

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Cilleruelo,The additive completion ofk-th powers,J

J. Cilleruelo,The additive completion ofk-th powers,J. Number Theory44(1993), 237–243

work page 1993

[9] [9]

Ding,Green’s problem on additive complements of the squares,C

Y. Ding,Green’s problem on additive complements of the squares,C. R. Math. Acad. Sci. Paris358 (2020), 897–900

work page 2020

[10] [10]

Ding, Y.-C

Y. Ding, Y.-C. Sun, L.-Y. Wang and Y. Xia,A note on additive complements of the squares, Discrete Math.349(2026), Paper 114763, 8 pp

work page 2026

[11] [11]

Donagi and M

R. Donagi and M. Herzog,On the additive completion of polynomial sets of integers,J. Number Theory3(1971), 150–154

work page 1971

[12] [12]

Erd˝ os,Problems and results in additive number theory,in: Colloque sur la Th´ eorie des Nombres, Bruxelles, 1955, George Thone, Li` ege Masson and Cie, Paris, 1956, pp

P. Erd˝ os,Problems and results in additive number theory,in: Colloque sur la Th´ eorie des Nombres, Bruxelles, 1955, George Thone, Li` ege Masson and Cie, Paris, 1956, pp. 127–137

work page 1955

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Erd˝ os,Problem 33,Proc

P. Erd˝ os,Problem 33,Proc. Number Theory Conf., Boulder, Colorado, 1963

work page 1963

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Habsieger,On the additive completion of polynomial sets,J

L. Habsieger,On the additive completion of polynomial sets,J. Number Theory51(1995), 130– 135

work page 1995

[15] [15]

Moser,On the Additive Completion of Sets of Integers,Proceedings of Symposia in Pure Math- ematics, vol

L. Moser,On the Additive Completion of Sets of Integers,Proceedings of Symposia in Pure Math- ematics, vol. VIII, Amer. Math. Soc., Providence, RI, 1965, pp. 175–180

work page 1965

[16] [16]

D. S. Ramana,Some Topics in Analytic Number Theory,PhD thesis, University of Madras, May 2000

work page 2000

[17] [17]

D. S. Ramana,A report on additive complements of the squares,in: Number Theory and Discrete Mathematics, Chandigarh, 2000, in: Trends Math., Birkh¨ auser, Basel, 2002, pp. 161–167

work page 2000

[18] [18]

I. Z. Ruzsa, Additive complements of lacunary sequences, Combinatorica,21(2001), 279–291

work page 2001

[19] [19]

I. Z. Ruzsa,On a problem of P. Erd˝ os,Canad. Math. Bull,15(1972), 309–310. (Yuchen Ding 1,2) 1School of Mathematics, Yangzhou University, Yangzhou 225002, People’s Republic of China 2HUN-REN Alfr´ed R´enyi Institute of Mathematics, Budapest, Pf. 127, H-1364 Hun- gary Email address:ycding@yzu.edu.cn (Zihan Zhang) School of Mathematical Sciences and LPMC, ...

work page 1972