Dense signed sums of non-integer powers

David Treeby

arxiv: 2606.06544 · v1 · pith:F6GVT3DInew · submitted 2026-06-04 · 🧮 math.GM

Dense signed sums of non-integer powers

David Treeby This is my paper

Pith reviewed 2026-06-27 22:54 UTC · model grok-4.3

classification 🧮 math.GM

keywords dense signed sumsnon-integer powersThue-Morse blocksProuhet-Tarry-Escott cancellationpartial sum densitypower sumssign choices

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The pith

For non-integer j>0, signed partial sums of k^j can be made dense in the reals.

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

The paper shows that when the exponent j is positive but not an integer, signs can be chosen for each term k^j so the running totals get arbitrarily close to any real number. It groups terms into Thue-Morse blocks whose cancellation creates small nonzero increments whose total size grows without bound. A steering choice of block signs then directs the sums to fill the line densely. A reader cares because this highlights how sign freedom interacts with the growth rate of powers, allowing control that may fail for integer exponents.

Core claim

If j>0 is not an integer, then there is a choice of signs ε_k∈{±1} such that the partial sums ∑_{k=1}^N ε_k k^j are dense in R. The proof groups consecutive terms into Thue-Morse blocks, whose Prouhet-Tarry-Escott cancellation produces nonzero block sums tending to zero but with divergent total variation. A standard steering argument then chooses block signs so that the resulting partial sums visit arbitrarily small neighbourhoods of every real number.

What carries the argument

Thue-Morse blocks with Prouhet-Tarry-Escott cancellation, which produce small nonzero block sums whose divergent total variation permits steering the partial-sum path.

If this is right

The partial sums enter every open interval on the real line.
The construction works for every non-integer positive exponent.
Block-wise sign choice suffices to achieve the density once the block sums satisfy the zero-limit and divergent-variation conditions.
The result separates the non-integer case from integer exponents where algebraic identities may prevent such density.

Where Pith is reading between the lines

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

The same block-steering technique might apply to other slowly growing sequences whose partial sums lack polynomial closed forms.
Integer exponents could produce bounded gaps or recurrence relations that block density for any sign choice.
Numerical checks for small non-integers such as j=1/2 could locate explicit sign patterns that keep sums within shrinking target intervals.
The divergence of block-sum variation may link to questions about the discrepancy of signed power sums in other metrics.

Load-bearing premise

The Prouhet-Tarry-Escott cancellation inside Thue-Morse blocks for non-integer j produces nonzero block sums that tend to zero while their total variation diverges.

What would settle it

An explicit non-integer j together with a sign sequence (or proof that none exists) whose partial sums remain outside some open interval of positive length for all N.

read the original abstract

We prove that if $j>0$ is not an integer, then there is a choice of signs $\varepsilon_k\in\{\pm1\}$ such that the partial sums $ \sum_{k=1}^{N}\varepsilon_k k^j $ are dense in $\mathbb R$. The proof groups consecutive terms into Thue--Morse blocks, whose Prouhet--Tarry--Escott cancellation produces nonzero block sums tending to zero but with divergent total variation. A standard steering argument then chooses block signs so that the resulting partial sums visit arbitrarily small neighbourhoods of every real number.

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

The paper gives a working existence proof for density of signed k^j sums when j is non-integer, built on Thue-Morse blocks plus steering, with no load-bearing gaps.

read the letter

The central result is that for non-integer j > 0 there exist signs making the partial sums of k^j dense in R. The proof groups terms into fixed-length Thue-Morse blocks sized to 2^n with n = floor(j) + 1, applies Prouhet-Tarry-Escott cancellation to kill powers through degree n-1, and obtains block sums that decay like c m^{j-n} with c nonzero exactly because j is not integer. Those blocks go to zero while their absolute sums diverge, so the standard steering construction produces density.

The paper does this cleanly with classical tools and no extra machinery. The key step that the leading coefficient survives only for non-integer j is handled directly from the binomial expansion of the block sums, and the divergence of total variation follows from the exponent lying in (0,1). That part is explicit and does not rely on fitting or hidden constants.

Soft spots are small. The steering argument is the usual one, so it adds no new difficulty. One could ask for explicit rates or uniformity in j, but the existence claim does not need them. The block construction is deterministic once the pattern is fixed, which keeps the argument self-contained.

This note is aimed at people working on signed sums, discrepancy, or analytic number theory. It supplies a reusable construction rather than resolving a long-standing problem. The reasoning is coherent on its own terms, so the paper deserves a serious referee even if the result is narrow.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that if j>0 is not an integer, then there exists a choice of signs ε_k ∈ {±1} such that the partial sums ∑_{k=1}^N ε_k k^j are dense in R. The argument partitions the sum into fixed-length Thue-Morse blocks of size 2^n (n = floor(j)+1), invokes the Prouhet-Tarry-Escott cancellation property to annihilate powers 0 through n-1, obtains block sums s_l that are asymptotically nonzero and behave as c N_l^{j-n} (hence s_l → 0 while ∑ |s_l| diverges because n-j ∈ (0,1)), and applies a standard steering construction on these increments to produce density of the block-end partial sums (and therefore of all partial sums).

Significance. If correct, the result supplies an explicit combinatorial construction, based on the Thue-Morse sequence and Prouhet-Tarry-Escott solutions, that realizes density for signed power sums precisely when the exponent is non-integral. The proof is self-contained, uses only standard combinatorial objects, and isolates the non-integrality of j as the sole source of the nonzero leading coefficient in the block-sum asymptotics. This strengthens the literature on discrepancy and signed sums by giving a uniform method that works for all non-integer j>0.

minor comments (2)

[Abstract] The abstract would be strengthened by a single sentence stating the key asymptotic form of the block sums and the reason the constant is nonzero.
A brief remark comparing the construction to the integer-exponent case (where the same block sums vanish identically) would help readers situate the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, the accurate summary of the main result, and the positive assessment of its significance. We are grateful for the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation is a direct existence proof that groups terms into fixed-length Thue-Morse blocks, invokes the standard Prouhet-Tarry-Escott cancellation property (which holds for the chosen block size when j is non-integer), obtains block sums s_l ~ c N_l^{j-n} with c nonzero, and applies a standard steering construction on the resulting increments. No equations reduce a claimed result to its own inputs by construction, no parameters are fitted to data, and the load-bearing steps rely on externally known combinatorial facts rather than self-citation chains or ansatzes imported from the authors' prior work. The paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on two domain assumptions about the behavior of Thue-Morse blocks for non-integer exponents and the applicability of a standard steering construction; no free parameters or new entities are introduced.

axioms (2)

domain assumption Thue-Morse blocks via Prouhet-Tarry-Escott produce nonzero sums tending to zero with divergent total variation when the exponent is non-integer.
Invoked to obtain the block sums that the steering argument then controls.
domain assumption A standard steering argument applied to such block sums yields density in R.
Used to conclude that the partial sums visit every neighborhood.

pith-pipeline@v0.9.1-grok · 5605 in / 1408 out tokens · 30533 ms · 2026-06-27T22:54:28.648437+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

M. N. Bleicher, On Prielipp’s problem on signed sums ofkth powers,Journal of Number Theory56(1996), no. 1, 36–51. doi:10.1006/jnth.1996.0004

work page doi:10.1006/jnth.1996.0004 1996
[2]

H. B. Yu, Signed sums of polynomial values,Proceedings of the American Mathematical Society130(2002), no. 6, 1623–1627. doi:10.1090/S0002-9939-01-06461-9

work page doi:10.1090/s0002-9939-01-06461-9 2002
[4]

Allouche and J

J.-P. Allouche and J. Shallit, The ubiquitous Prouhet–Thue–Morse sequence, inSequences and Their Applications, Springer Series in Discrete Mathematics and Theoretical Computer Science, Springer, London, 1999, pp. 1–16. doi:10.1080/00029890.2024.2419799. 6

work page doi:10.1080/00029890.2024.2419799 1999

[1] [1]

M. N. Bleicher, On Prielipp’s problem on signed sums ofkth powers,Journal of Number Theory56(1996), no. 1, 36–51. doi:10.1006/jnth.1996.0004

work page doi:10.1006/jnth.1996.0004 1996

[2] [2]

H. B. Yu, Signed sums of polynomial values,Proceedings of the American Mathematical Society130(2002), no. 6, 1623–1627. doi:10.1090/S0002-9939-01-06461-9

work page doi:10.1090/s0002-9939-01-06461-9 2002

[3] [4]

Allouche and J

J.-P. Allouche and J. Shallit, The ubiquitous Prouhet–Thue–Morse sequence, inSequences and Their Applications, Springer Series in Discrete Mathematics and Theoretical Computer Science, Springer, London, 1999, pp. 1–16. doi:10.1080/00029890.2024.2419799. 6

work page doi:10.1080/00029890.2024.2419799 1999