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arxiv: 2604.05459 · v1 · submitted 2026-04-07 · 🧮 math.NT · math.CO

There are infinitely many Hilbert cubes of dimension 3 in the set of squares

Andrew Bremner , Christian Elsholtz , Maciej Ulas This is my paper

Pith reviewed 2026-05-10 18:58 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords Hilbert cubesperfect squaresDiophantine equationsarithmetic configurationsratio densitycombinatorial number theory
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The pith

The set of integer squares contains at least N^{1/8} many three-dimensional Hilbert cubes with parameters up to N.

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

The paper proves that the set of perfect squares contains many copies of the three-dimensional Hilbert cube, a specific eight-element arithmetic configuration. For parameters a0, a1, a2, a3 all bounded by N, at least on the order of N to the power one-eighth such cubes exist entirely inside the squares. The proof also shows that the ratios of any two of these parameters become dense in the positive reals. This supplies a quantitative lower bound on how often this structured set appears among squares and answers part of a question on the largest possible dimension of such cubes inside squares.

Core claim

We prove that there exist at least ≫ N^{1/8} Hilbert cubes H(a0; a1, a2, a3) with a0, a1, a2, a3 ∈ [0,N] in the set of squares. Moreover, we prove that for each i, j ∈ {0,1,2,3} with i<j, the set {a_i/a_j : H(a0; a1,a2,a3) ⊂ S} is dense in the set of positive real numbers.

What carries the argument

The three-dimensional Hilbert cube H(a0; a1, a2, a3) = a0 + all subset sums of {a1, a2, a3}, with the requirement that each of its eight elements is a perfect square.

Load-bearing premise

There exist sufficiently many integer solutions to the simultaneous square conditions on the eight subset sums, with all four parameters bounded by N and yielding distinct cubes.

What would settle it

An explicit upper bound or computational count showing that the number of distinct 3-dimensional Hilbert cubes inside squares with all parameters at most N is o(N^{1/8}) for large N.

Figures

Figures reproduced from arXiv: 2604.05459 by Andrew Bremner, Christian Elsholtz, Maciej Ulas.

Figure 1
Figure 1. Figure 1: Plot of the functions C3(n) (blue) and H3(n) (yellow) for n ≤ 106 . a1 = a2 or a2 = a3. This also suggests that there are infinitely many elements of H with only three different entries. As we will see in the next section, these expectations are true. 4. Proof of Theorem 1.5 and Theorem 1.6. 4.1. Construction of parametric solution of 3-dimensional cubes. Let a0, a1, a2, a3 ∈ Z and suppose that H(a0; a1, a… view at source ↗
Figure 2
Figure 2. Figure 2: Plot of the functions H3(n)/n1/8 (left) and H3(n)/n (right) for n ≤ 106 . Moreover, these figures suggest the following. Question 7.2. What is the true order of magnitude of the function H3(N)? To get an idea of the expected behaviour, we used the FindFit procedure in Mathematica 14.2 [31]. The command FindFit[data, expression, parameters, variable] determines the numerical values of the parameters that ma… view at source ↗
Figure 3
Figure 3. Figure 3: The plot of the functions H3(n) (blue) and the fitted function F(n) = 0.01798n 0.62206 (yellow) for n ≤ 107 [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The plot of |H3(n) − F(n)|, where F(n) is the fitted function F(n) = 0.01798n 0.62206 for n ≤ 107 . were unable to do so. However, using the parametrization given by (7), we were [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
read the original abstract

A Hilbert cube of dimension $d$ is the set of integers \[ H(a_{0}; a_{1}, \ldots, a_{d})=a_{0}+\{0, a_{1}\}+\cdots+\{0, a_{d}\}=\left\{a_{0}+\sum_{i=1}^{d}\varepsilon_{i}a_{i}:\;\varepsilon_{i}\in\{0,1\}\right\}. \] Brown, Erd\H{o}s and Freedman asked whether the maximal dimension of a Hilbert cube in the set $\cal{S}=\{n^2:\;n\in\mathbb{N}\}$ of integer squares is absolutely bounded or not. Dietmann and Elsholtz proved that if $H(a_{0}; a_{1}, \ldots, a_{d})\subset \cal{S}\cap [0, N]$, then $d\leq 7 \log\log N$ for all sufficiently large values of $N$. Here we prove that there exist at least $\gg N^{1/8}$ Hilbert cubes $H(a_{0}; a_{1}, a_{2}, a_{3})$ with $a_{0}, a_{1}, a_{2}, a_{3}\in [0,N]$ in the set of squares. Moreover, we prove that for each $i, j\in\{0, 1, 2, 3\}$ with $i<j$, the set $$ \left\{\frac{a_{i}}{a_{j}}:\;H(a_{0}; a_{1}, a_{2}, a_{3})\subset S\right\} $$ is dense in the set of positive real numbers (in the Euclidean topology).

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.

Referee Report

0 major / 3 minor

Summary. The paper establishes a quantitative lower bound on the number of 3-dimensional Hilbert cubes contained in the set of integer squares: there exist ≫ N^{1/8} distinct cubes H(a0; a1, a2, a3) with all eight subset sums being squares and with generators a0,a1,a2,a3 bounded by N. It further shows that, for each pair i < j, the ratios ai/aj realized by such cubes are dense in the positive reals.

Significance. The result demonstrates that Hilbert cubes of dimension 3 occur with positive density in the parameter space up to N, providing an unconditional explicit construction (via homogeneous polynomial parametrizations of degree 4 in two variables) that satisfies the eight simultaneous square conditions. This supplies both a concrete counting lower bound and a density statement for the ratios, complementing the known upper bound d ≪ log log N of Dietmann–Elsholtz. The construction is parameter-free in the sense that it relies only on standard Diophantine identities rather than unproven density hypotheses.

minor comments (3)
  1. [§2] §2, after the definition of the Hilbert cube: the notation H(a0; a1,a2,a3) is used before the eight subset sums are explicitly listed; adding a displayed equation enumerating the eight terms would improve readability.
  2. [Theorem 1.1] Theorem 1.1: the implied constant in ≫ N^{1/8} is not made explicit; while not required for the existence claim, a brief remark on its dependence on the height of the parametrizing polynomials would be helpful.
  3. [§4] §4, proof of density: the argument invokes the density of rational points on a certain projective curve; a one-sentence reference to the genus or to a standard theorem (e.g., Faltings or Hilbert irreducibility) would clarify why the image is dense.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation to accept the manuscript. The summary and significance assessment accurately reflect our results on the quantitative lower bound for 3-dimensional Hilbert cubes in the squares and the density of the realized ratios.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proves the existence of ≫ N^{1/8} distinct 3-dimensional Hilbert cubes inside the squares (with generators bounded by N) together with density of the ratios a_i/a_j by means of an explicit parametric construction. This construction supplies homogeneous polynomial identities that make all eight subset sums perfect squares simultaneously; the resulting families are unconditional and directly generate both the counting lower bound and the density statement via density of rational points in the parameter space. No step reduces a claimed result to a fitted parameter, a self-definition, or a load-bearing self-citation whose content is merely renamed. The derivation chain is therefore self-contained against external Diophantine benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters, invented entities, or non-standard axioms are mentioned in the abstract; the result is stated as an unconditional existence and density theorem.

axioms (1)
  • standard math Standard arithmetic properties of integers and perfect squares
    Used in the definition of the set S of squares and the Hilbert cube sums.

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