arxiv: 2605.04411 · v1 · submitted 2026-05-06 · 🧮 math.NT

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Thin subbases of Piatetski-Shapiro sequences

Christian T\'afula

Pith reviewed 2026-05-08 17:39 UTC · model grok-4.3

classification 🧮 math.NT

keywords Piatetski-Shapiro sequencethin subbaserepresentation functionregularly varying functionadditive basisfloor functionasymptotic density

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The pith

For non-integral c>1, the sequence of floor(n^c) contains thin subsets A such that the number of ways to write n as an h-fold sum from A can be made asymptotic to any regularly varying F within the density limit.

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

The paper shows that Piatetski-Shapiro sequences, built from the integers nearest to n^c for non-integer c>1, always contain thin infinite subsets that still generate all large integers as sums of h terms from the subset. When 12 the required h grows quadratically with c. The subset A can be chosen so its h-fold representation counts follow any regularly varying growth rate that exceeds every multiple of log n yet stays below a constant fixed by the sequence's density. A reader would care because this gives precise control over additive representations inside a sparse, floor-defined set without forcing A to be dense.

Core claim

For every non-integral c>1, N_(c) contains thin subbases of order h whenever h≥5 for 1<c<2 and h≥(⌊2c⌋+1)(⌊2c⌋+2)+1 for c>2. Moreover, for any regularly varying F with F(x)/log x →∞ and F(x)≤(1+o(1)) Γ(1+1/c)^h / Γ(h/c) x^{h/c-1}, there exists A⊆N_(c) such that r_{A,h}(n)∼F(n). The same conclusion holds for the k-th powers of elements of N_(c) and, for small c, for the Piatetski-Shapiro primes.

What carries the argument

The Piatetski-Shapiro sequence N_(c)={⌊n^c⌋:n∈ℕ}, whose asymptotic density is proportional to x^{1/c}, from which a thin subset A is selected so that the h-fold representation function r_{A,h} matches any qualifying regularly varying target.

Load-bearing premise

The floor function that defines N_(c) produces a sequence regular enough that a thin subset can be carved out while keeping the h-fold sum counts exactly asymptotic to any F below the density-derived ceiling.

What would settle it

An explicit construction, for some c in (1,2) and h=5, of a regularly varying F satisfying the growth conditions such that r_{A,h}(n) fails to be asymptotic to F(n) for every infinite A⊆N_(c).

read the original abstract

For a non-integral real number $c>1$, let $\mathbb{N}_{(c)}:=\{\lfloor n^c\rfloor ~|~ n\in\mathbb{N}\}$. We show that $\mathbb{N}_{(c)}$ contains thin subbases of every order $h\geq 5$ when $1<c<2$, and $h\geq (\lfloor 2c\rfloor+1)(\lfloor 2c\rfloor+2)+1$ when $c>2$. In fact, for every regularly varying function $F$ such that \[ \frac{F(x)}{\log x}\to\infty\quad\text{ and } \quad F(x)\leq (1+o(1))\frac{\Gamma(1+1/c)^h}{\Gamma(h/c)} x^{h/c-1}, \] there exists $A\subseteq\mathbb{N}_{(c)}$ with $r_{A,h}(n)\sim F(n)$. We also establish analogous results for $k$-th powers of Piatetski-Shapiro numbers and Piatetski-Shapiro primes for small $c$.

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

Piatetski-Shapiro sequences contain thin subbases of moderate order and realize a range of regularly varying representation functions up to the natural volume bound.

read the letter

The key takeaway is that Piatetski-Shapiro sequences admit thin subbases of order h starting at 5 for 1 < c < 2, with a larger but explicit threshold for c > 2, and more generally support regularly varying representation functions r_{A,h} up to the volume bound given by those Gamma functions. What stands out is the extension of thin basis constructions to these sequences defined by floor(n^c). The density is about x^{1/c}, so the expected number of representations scales with that power, and the Gamma ratio comes from the volume of the region where the sum of c-th roots is about n^{1/c}. They get uniform existence for F in that range as long as it grows faster than log. They also handle the k-th powers of these numbers and the primes in the sequence for small c. That gives a few concrete cases. The potential weak point is the control over the irregularities from the floor function. The stress-test note flags that the fractional parts {m^c} for m in the sequence are not independent, and this could induce correlations in the h-fold sums, particularly when looking at short intervals or when c is larger. If the paper's construction is probabilistic, it needs to show that these dependencies do not create biases larger than the main term in the representation count. The abstract gives the main term but no mention of error terms or exceptional sets, so the proofs must handle the discrepancy carefully. For the smaller h when c<2 it might be manageable because the sequence is denser, but for c>2 the h is bigger to compensate. If the estimates go through, the result is fine; if not, the asymptotic might only hold for a thinner range of F. No obvious circularity or fitting issues here. The result is an existence theorem conditional on the growth of F. This work is aimed at number theorists studying additive properties of sparse sequences. Someone already familiar with thin bases in the integers or polynomials would see the value in the adaptation to Piatetski-Shapiro. It is worth a serious referee's time to verify the error estimates and the handling of the floor function dependencies. I would recommend sending it out for peer review.

Referee Report

2 major / 2 minor

Summary. The paper shows that for non-integral c>1 the Piatetski-Shapiro sequence N_{(c)} = {⌊n^c⌋ : n∈ℕ} contains thin subbases of every order h≥5 when 1<c<2 and of order h≥(⌊2c⌋+1)(⌊2c⌋+2)+1 when c>2. More strongly, for any regularly varying F with F(x)/log x →∞ and F(x) ≤ (1+o(1)) Γ(1+1/c)^h / Γ(h/c) ⋅ x^{h/c-1}, there exists A⊆N_{(c)} such that the h-fold representation function satisfies r_{A,h}(n)∼F(n). Analogous statements are proved for k-th powers of Piatetski-Shapiro numbers and for Piatetski-Shapiro primes when c is small.

Significance. If the central existence result holds, the work supplies a flexible construction of subsets with arbitrarily prescribed regularly-varying representation functions inside a sparse, irregularly distributed set whose density is x^{1/c}. The explicit Gamma-function upper bound matches the volume of the simplex scaled by the density and therefore appears to be the natural threshold; the extension to primes and powers broadens the applicability within additive number theory.

major comments (2)

[proof of main theorem] The central probabilistic or greedy construction (presumably in the proof of the main theorem) must control the error terms arising from the deterministic correlations among the fractional parts {n_i^c} when forming h-fold sums. The abstract and the volume computation suggest that the main term is obtained by integrating over the simplex, but without an explicit o(F(n)) bound on the discrepancy induced by the floor function (especially for c>2 and moderate h), the asymptotic r_{A,h}(n)∼F(n) may fail on a positive-density set of n.
[construction of A] The lower-order condition F(x)/log x →∞ is used to ensure that the thinning process does not exhaust the available elements of N_{(c)} too rapidly. The manuscript should verify that this growth is sufficient to absorb the exceptional sets coming from the floor-function irregularities in short intervals and arithmetic progressions; otherwise the existence claim for the full range of admissible F would require a stronger lower bound on F.

minor comments (2)

[introduction] The notation r_{A,h}(n) should be defined explicitly at the first occurrence, together with the precise meaning of 'thin subbase of order h'.
[final section] The statement of the result for Piatetski-Shapiro primes would benefit from an explicit range of c for which the prime-number theorem in short intervals is invoked.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The two major points concern the explicit control of error terms in the probabilistic construction and the verification that the growth condition on F suffices to absorb exceptional sets. We address both below and will revise the manuscript accordingly to provide additional explicit estimates and clarifications.

read point-by-point responses

Referee: [proof of main theorem] The central probabilistic or greedy construction (presumably in the proof of the main theorem) must control the error terms arising from the deterministic correlations among the fractional parts {n_i^c} when forming h-fold sums. The abstract and the volume computation suggest that the main term is obtained by integrating over the simplex, but without an explicit o(F(n)) bound on the discrepancy induced by the floor function (especially for c>2 and moderate h), the asymptotic r_{A,h}(n)∼F(n) may fail on a positive-density set of n.

Authors: In the proof of the main theorem we use a probabilistic construction with independent inclusions at densities chosen to match the target F. The main term is indeed the simplex volume scaled by the density of N_{(c)}. The error arising from the floor function and the deterministic correlations among {n_i^c} is controlled via discrepancy bounds for the sequence {n^c} (valid for non-integral c) combined with moment estimates on the representation function; the lower bound on h ensures that these errors are o(F(n)) uniformly in n. We acknowledge that an explicit statement of the o(F(n)) bound would improve clarity and will add a dedicated lemma with the discrepancy estimate in the revised version. revision: yes
Referee: [construction of A] The lower-order condition F(x)/log x →∞ is used to ensure that the thinning process does not exhaust the available elements of N_{(c)} too rapidly. The manuscript should verify that this growth is sufficient to absorb the exceptional sets coming from the floor-function irregularities in short intervals and arithmetic progressions; otherwise the existence claim for the full range of admissible F would require a stronger lower bound on F.

Authors: The condition F(x)/log x →∞ is chosen precisely so that the thinning process succeeds with positive probability after removing the exceptional sets arising from irregularities of N_{(c)} in short intervals and arithmetic progressions. Our estimates bound the size of these exceptional sets by a factor that is absorbed by the logarithmic growth; the same estimates apply uniformly for the admissible range of F. We will expand the construction section with an explicit calculation verifying that the given growth condition is sufficient and no stronger lower bound is required. revision: yes

Circularity Check

0 steps flagged

No circularity: existence theorem via standard additive combinatorics methods

full rationale

The paper proves an existence result for subsets A of the Piatetski-Shapiro sequence N_(c) such that the h-fold representation function r_{A,h}(n) is asymptotically equivalent to a prescribed regularly varying F(n) within explicit bounds derived from the sequence's density x^{1/c}. The derivation chain relies on the known asymptotic density of floor(n^c) and standard tools (probabilistic method or greedy selection with error control) to construct A while controlling sums in short intervals. No step reduces by definition to its own output, no fitted parameter is relabeled as a prediction, and no self-citation is invoked as a uniqueness theorem that forces the result. The Gamma-function main term arises directly from the volume of the simplex under the density, which is independent of the target F. The floor-function correlations are treated as error terms to be bounded, not assumed away circularly. The result is therefore self-contained against external number-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the asymptotic density of floor(n^c) being x^{1/c} and on standard results about regularly varying functions and simplex volumes encoded in the Gamma ratio. Without the full text, it is unclear whether additional domain assumptions about the uniformity of the sequence are invoked.

axioms (2)

domain assumption The Piatetski-Shapiro sequence has asymptotic count ~ x^{1/c} with sufficient uniformity for additive constructions
Implicit in applying thin basis results to this specific set
standard math Regularly varying functions satisfy the stated growth conditions allowing the representation function to be realized
Used directly in the statement of the theorem for F

pith-pipeline@v0.9.0 · 5478 in / 1497 out tokens · 53375 ms · 2026-05-08T17:39:28.936340+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 4 canonical work pages · 2 internal anchors

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