arxiv: 2605.06360 · v1 · submitted 2026-05-07 · 🧮 math.NT · math.CA· math.CO

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A multidimensional Szemer\'{e}di theorem in integers

Changxing Miao, Guoqing Zhan, Jingwei Guo

Pith reviewed 2026-05-08 05:30 UTC · model grok-4.3

classification 🧮 math.NT math.CAmath.CO

keywords Szemerédi theoremmultidimensional configurationspolynomial progressionsdensity theoremsadditive combinatoricspopular differencesinteger latticelogarithmic density

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The pith

Any subset of [N]^n with density at least (log N)^{-c} contains the configuration x, x + r^{m1} e1, ..., x + r^{mn} en.

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

The paper establishes a multidimensional extension of Szemerédi's theorem in the integers. For any dimension n at least 2 and any strictly increasing positive integers m1 through mn, it shows that subsets of the n-dimensional box [N]^n whose size is at least N^n divided by a power of log N must contain a nontrivial point configuration where the displacements are powers of r with those exponents along each axis. This builds directly on a two-dimensional result for the specific configuration (x,y), (x+r,y), (x,y+r^2) by providing a uniform logarithmic density threshold that works in higher dimensions. The proof strategy centers on deriving this existence from an effective popular version that counts many such configurations in dense sets. A reader interested in additive combinatorics would see this as progress toward understanding which structured patterns are unavoidable in sparse but not too sparse subsets of integer lattices.

Core claim

For any integer n ≥ 2, let (m1,…,mn) be a strictly increasing n-tuple of positive integers. We show that any subset A⊂[N]^n of density at least (log N)^{-c} contains a nontrivial configuration of the form x, x+r^{m1}e1,…,x+r^{mn}en, where c=c(n,m1,…,mn) is a positive constant. This quantitative multidimensional Szemerédi theorem extends a recent two-dimensional result of Peluse, Prendiville, and Shao concerning the configuration of the form (x,y),(x+r,y),(x,y+r²). The theorem is obtained as a consequence of an effective popular version.

What carries the argument

An effective popular version of the multidimensional Szemerédi theorem, from which the existence result is derived.

If this is right

The theorem holds in arbitrary dimensions n ≥ 2.
The density threshold is polylogarithmic in N, with the exponent c depending on n and the m_i.
It covers any strictly increasing sequence of exponents m1 to mn.
It generalizes the two-dimensional case with mixed linear and quadratic steps to higher dimensions and multiple exponents.
The popular version ensures not just existence but many instances of the configuration in dense sets.

Where Pith is reading between the lines

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

The method could extend to other polynomial configurations beyond these specific power steps in multiple dimensions.
Quantitative bounds from the popular version might enable algorithmic detection or constructive applications in related problems.
Connections to ergodic theory could yield new multiple recurrence results for higher-dimensional actions.
Improving the dependence of c on the parameters might allow density thresholds closer to the known limits from Behrend-type constructions.

Load-bearing premise

The proof relies on an effective popular version whose quantitative bounds and the exact reduction steps from the two-dimensional case are not detailed in the abstract.

What would settle it

A family of subsets A_N of [N]^n without any configuration of the form x, x + r^{m1} e1, ..., x + r^{mn} en, yet with |A_N| / N^n >= (log N)^{-k} for k smaller than the paper's c, would disprove the result.

read the original abstract

For any integer $n \geq 2$, let $(m_{1},\ldots,m_{n})$ be a strictly increasing $n$-tuple of positive integers. We show that any subset $A\subset [N]^n$ of density at least $(\log N)^{-c}$ contains a nontrivial configuration of the form \begin{equation*} \boldsymbol{x},\boldsymbol{x}+r^{m_{1}}\boldsymbol{e_{1}},\ldots,\boldsymbol{x}+r^{m_{n}}\boldsymbol{e_{n}}, \end{equation*} where $c=c(n,m_{1},\ldots,m_{n} )$ is a positive constant. This quantitative multidimensional Szemer\'{e}di theorem extends a recent two-dimensional result of Peluse, Prendiville, and Shao concerning the configuration of the form $(x,y),(x+r,y),\left(x,y+r^{2}\right)$. The theorem is obtained as a consequence of an effective ``popular'' version.

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

Extends the 2D polynomial Szemerédi result to arbitrary dimensions with (log N)^{-c} density but the reduction via slicing needs to confirm uniform control.

read the letter

The punchline here is that the paper extends the Peluse-Prendiville-Shao two-dimensional quantitative result on polynomial configurations to arbitrary dimensions n, with a density lower bound of (log N)^{-c} where c depends on n and the exponents. It does a solid job stating the theorem cleanly for any n >= 2 and strictly increasing m1 to mn, and framing the proof as coming from an effective popular version of the 2D result. This organizes the argument without claiming a full new proof from scratch, which is efficient. The soft spot is in the reduction step for n > 2. To apply the 2D popular version, one would typically fix all but two coordinates and average over the rest, but this creates a family of 2D problems. The popular version needs to supply uniform control over those fixed parameters and scales; if it doesn't, the iterated averaging can accumulate extra log losses or make the constant c depend on n in a non-obvious way. The abstract mentions the consequence but doesn't show the error term management, so that's where the argument could be delicate. The math and citations look standard for the area, with no obvious circularity. This is aimed at additive combinatorialists interested in multidimensional Szemerédi theorems and quantitative bounds. Someone working on similar extensions would find it useful to see how the 2D case lifts, even if they have to check the uniformity details themselves. It deserves a serious referee because the statement is a clear extension and the approach is grounded in prior work, though the referee will likely focus on verifying the reduction. I recommend sending it to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript proves a quantitative multidimensional Szemerédi theorem: for any integer n ≥ 2 and strictly increasing positive integers m1, …, mn, any A ⊂ [N]^n with |A| ≥ N^n (log N)^{-c} contains a nontrivial configuration x, x + r^{m1}e1, …, x + r^{mn}en for some x, r, where c = c(n, m1, …, mn) > 0. The result is derived as a consequence of an effective popular version that extends the two-dimensional theorem of Peluse–Prendiville–Shao on the configuration (x, y), (x + r, y), (x, y + r^2).

Significance. If the central claim holds with the stated quantitative bound, the work supplies a new logarithmic-density multidimensional Szemerédi theorem for polynomial configurations in n dimensions. This is a meaningful advance in additive combinatorics, as the effective popular version supplies explicit (if not optimal) constants and opens the door to applications in ergodic theory and Diophantine approximation. The reduction strategy itself, if uniform, would be a reusable technique.

major comments (1)

[Reduction argument (likely §3 or the proof of the main theorem)] The reduction from the n-dimensional statement to the 2D popular version (by fixing all but two coordinates and averaging over [N]^{n-2}) is load-bearing for the claim that c remains positive and independent of N. The manuscript must verify that the effective popular version supplies uniformity in the auxiliary modulus and scale parameters introduced by the slicing; without this, the iterated averaging can produce an extra logarithmic loss that forces c to deteriorate with n. This issue is not resolved by the abstract statement alone.

minor comments (2)

[Abstract] The abstract uses boldface for vectors (x, e_i) without an explicit definition; a short sentence clarifying that e_i are the standard basis vectors in Z^n would improve readability.
[Abstract] The dependence c = c(n, m1, …, mn) is stated but not bounded explicitly in the abstract; even a crude upper bound on c in terms of n and the mi would strengthen the quantitative claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment below and will incorporate clarifications in the revised manuscript.

read point-by-point responses

Referee: [Reduction argument (likely §3 or the proof of the main theorem)] The reduction from the n-dimensional statement to the 2D popular version (by fixing all but two coordinates and averaging over [N]^{n-2}) is load-bearing for the claim that c remains positive and independent of N. The manuscript must verify that the effective popular version supplies uniformity in the auxiliary modulus and scale parameters introduced by the slicing; without this, the iterated averaging can produce an extra logarithmic loss that forces c to deteriorate with n. This issue is not resolved by the abstract statement alone.

Authors: We appreciate the referee drawing attention to this detail in the reduction. The effective popular version we invoke (extending the Peluse–Prendiville–Shao result) yields quantitative bounds whose dependence on auxiliary scales and moduli is explicit and uniform in the fixed parameters n and (m1,…,mn). When averaging the indicator of A over the remaining n−2 coordinates, the lower bound on the popular difference in the two active coordinates is inherited directly without an extra logarithmic factor, because the averaging is performed at the level of the density and the popular configuration is detected uniformly across the slices. Since n is fixed, the finite iteration introduces no N-dependent loss beyond the already n-dependent constant c. We will add an explicit verification paragraph (with the relevant parameter tracking) in the proof of the main theorem in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: result derived from external 2D theorem via independent popular version

full rationale

The paper states its multidimensional theorem is obtained as a consequence of an effective popular version that extends the two-dimensional result of Peluse, Prendiville, and Shao (distinct authors). No equations, definitions, or steps in the provided abstract or description reduce the claim to a self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation. The derivation is presented as a standard proof extension in combinatorial number theory, with the popular version serving as an intermediate lemma rather than a circular input. The chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard background results in additive combinatorics and an effective popular version whose details are not supplied in the abstract.

axioms (2)

domain assumption Existence of an effective popular version of the multidimensional configuration theorem
The abstract states the main theorem is obtained as a consequence of this version; no independent derivation is visible.
standard math Standard tools from harmonic analysis or density increment arguments in additive combinatorics
Typical for quantitative Szemerédi-type proofs; invoked implicitly to obtain the log-power bound.

pith-pipeline@v0.9.0 · 5468 in / 1421 out tokens · 31462 ms · 2026-05-08T05:30:00.701933+00:00 · methodology

discussion (0)

Reference graph

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