arxiv: 2604.05768 · v1 · submitted 2026-04-07 · 🧮 math.DS · math.CO

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On the Furstenberg-Katznelson constant for the IP Szemeredi theorem over finite fields

Or Shalom

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Pith reviewed 2026-05-10 19:09 UTC · model grok-4.3

classification 🧮 math.DS math.CO

keywords IP Szemeredi theoremFurstenberg-Katznelson constantfinite fieldsergodic theorymultiple averagesRoth theoremarithmetic progressionscharacteristic factors

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The pith

Positive constants exist for the density of IP arithmetic progressions in finite vector spaces over finite fields.

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

In the integers, the Furstenberg-Katznelson constant c_IP guarantees a positive lower bound on the density of k-term arithmetic progressions within any set of density at least delta, with the differences forming an IP* set. This paper establishes that analogous positive constants exist when the ambient group is a vector space over a finite field. The result is qualitative in general but becomes quantitative with explicit bounds in the cases of Roth's theorem and the IP-Roth theorem. The approach uses ergodic theory to study the limits of multiple averages along IP sequences.

Core claim

The central claim is that the ergodic recurrence constants c^rec and c_IP^rec are positive for every k and delta in the setting of finite-dimensional vector spaces over finite fields. This follows from the existence of characteristic factors that govern the limit of the IP multiple ergodic averages, ensuring that the average of the product of indicators is at least some positive number depending only on k and delta whenever the density is at least delta. In the special cases corresponding to three-term progressions, the paper derives strong lower bounds on these constants.

What carries the argument

The characteristic factors for multiple ergodic averages along IP sets in vector spaces over finite fields, which determine the asymptotic behavior and positivity of the recurrence densities.

Load-bearing premise

The characteristic factors and the limits of the multiple ergodic averages along IPs exist and behave analogously in vector spaces over finite fields as they do in the integers.

What would settle it

A concrete counterexample would be a subset of a finite vector space with density at least some delta but with the liminf of the IP multiple average equal to zero for the corresponding configuration, which would make the constant zero.

read the original abstract

Bergelson et al. observed that Furstenberg's proof of Szemeredi's theorem provides a positive lower bound on the density of arithmetic progressions in sets of positive density in the integers. Namely, for every $\delta\in(0,1]$ and every $k\in \mathbb{N}$, there exists a positive constant $c=c(k,\delta)>0$ such that $$\{n\in \mathbb{N} : d(E\cap (E-n)\cap\dots\cap (E-(k-1)n))>c(k,\delta)\} \neq \emptyset$$ whenever $d(E)\ge \delta$. Similarly, Furstenberg and Katznelson proved the IP Szemeredi theorem, establishing in particular the existence of a constant $c_{\mathrm{IP}}=c_{\mathrm{IP}}(k,\delta)>0$ such that $$\{n\in \mathbb{N} : d(E\cap (E-n)\cap\dots\cap (E-(k-1)n))>c_{\mathrm{IP}}(k,\delta)\}$$ is $\mathrm{IP}^*$ whenever $d(E)\ge \delta$. In this paper, we study analogues of $c$ and $c_{\mathrm{IP}}$ and their ergodic-theoretic counterparts, $c^{\mathrm{rec}}$ and $c_{\mathrm{IP}}^{\mathrm{rec}}$, for vector spaces over finite fields. We provide a qualitative result and in special cases such as Roth's theorem and the IP-Roth theorem, we also provide strong quantitative bounds for these constants. Our tools are primarily ergodic theoretic; we study the characteristic factors and limit of multiple ergodic averages along $\mathrm{IP}$s in vector spaces over finite fields.

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 defines finite-field versions of the Furstenberg-Katznelson constants c and c_IP, proves their qualitative existence via ergodic theory, and gives explicit quantitative bounds for the Roth and IP-Roth cases.

read the letter

The core advance is the introduction of these constants for vector spaces over finite fields, together with the transfer of the combinatorial statements to statements about liminf of IP multiple ergodic averages being bounded below by a positive number depending only on density and length. The paper also extracts concrete lower bounds in the Roth and IP-Roth settings, which is more than the integer literature usually supplies for the general case. That part is cleanly executed and gives the reader something usable right away in additive combinatorics over F_q^n. The ergodic setup for characteristic factors and IP limits is laid out explicitly, which is helpful for anyone who wants to push the same ideas further in positive characteristic. The quantitative bounds rest on the same characteristic-factor analysis that works in the integer case, and the paper appears to verify that the relevant nilfactors or inverse limits behave as expected here. The main soft spot is that the argument still depends on the IP convergence and factor theorems carrying over without new obstructions from the lack of a total order or from characteristic p. The abstract and the stress-test note both flag this as the load-bearing step, and while the paper claims to study these factors, any referee will want to see the precise place where the van der Corput or nilsequence arguments are adapted rather than simply invoked by analogy. If those adaptations are only sketched, the quantitative claims lose some force. This is a paper for people already working in ergodic theory or quantitative additive combinatorics over finite fields. A reader who needs explicit constants for density-increment arguments in F_q-vector spaces will get value from the Roth cases even before the general theory is fully settled. It is coherent on its own terms and engages the existing literature directly, so it should go to peer review rather than be desk-rejected.

Referee Report

1 major / 2 minor

Summary. The manuscript studies analogues of the Furstenberg-Katznelson constants c(k,δ) and c_IP(k,δ) (and their recurrence counterparts c^rec and c_IP^rec) for the IP Szemerédi theorem in finite-dimensional vector spaces over finite fields F_q. It establishes a qualitative existence result that these constants are positive whenever the underlying density δ > 0, and derives explicit quantitative lower bounds in the special cases of Roth's theorem (k=3) and the IP-Roth theorem by transferring the combinatorial statements to ergodic-theoretic statements about liminf of IP multiple ergodic averages being bounded below by a positive constant depending only on δ and k. The primary tools are the analysis of characteristic factors and the existence of limits for multiple ergodic averages along IPs in this setting.

Significance. If the transfer of the ergodic machinery holds, the work provides both qualitative existence and, in the Roth/IP-Roth cases, concrete quantitative bounds for density-increment statements over F_q^n. This extends the ergodic approach to Szemerédi-type theorems beyond the integers to positive-characteristic additive groups, which may inform quantitative additive combinatorics results in finite fields. The explicit quantitative bounds in the special cases constitute a concrete strength.

major comments (1)

[Sections deriving characteristic factors and IP limits (around the statements of Theorems 1.3 and 1.5)] The qualitative existence of c and c_IP (and the quantitative bounds for Roth/IP-Roth) rests on the claim that characteristic factors for IP multiple ergodic averages exist and are of the expected nilfactor form in the F_q-vector-space setting. The manuscript must explicitly verify that the van der Corput differencing argument and the IP-limit theorem carry over when the underlying additive group lacks a total order and has positive characteristic; potential obstructions (e.g., failure of certain nilsequence approximations) are not addressed in the sections deriving the characteristic factors.

minor comments (2)

[Introduction] Notation for the constants c^rec and c_IP^rec should be introduced with a clear comparison table to the integer-case constants c and c_IP to avoid reader confusion.
[Abstract and §1] The abstract claims 'strong quantitative bounds' for Roth and IP-Roth; the manuscript should state the explicit dependence on δ and q in the bounds rather than leaving it implicit.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed reading of the manuscript and for highlighting the need for greater clarity on the transfer of ergodic-theoretic tools to the finite-field setting. The comment identifies a legitimate point about explicit verification, which we will address by expanding the relevant sections. We respond to the major comment below.

read point-by-point responses

Referee: [Sections deriving characteristic factors and IP limits (around the statements of Theorems 1.3 and 1.5)] The qualitative existence of c and c_IP (and the quantitative bounds for Roth/IP-Roth) rests on the claim that characteristic factors for IP multiple ergodic averages exist and are of the expected nilfactor form in the F_q-vector-space setting. The manuscript must explicitly verify that the van der Corput differencing argument and the IP-limit theorem carry over when the underlying additive group lacks a total order and has positive characteristic; potential obstructions (e.g., failure of certain nilsequence approximations) are not addressed in the sections deriving the characteristic factors.

Authors: We agree that an explicit verification would improve the exposition. The van der Corput differencing argument and the existence of IP-limits rely only on the abelian group structure and the combinatorial properties of IP sets; these hold verbatim in any abelian group, including (F_q^n, +), without requiring a total order. The identification of characteristic factors as nilfactors proceeds via the same PET-induction scheme used in the integer case. Positive characteristic introduces no obstructions for the relevant equidistribution or approximation results on nilsequences, because the underlying polynomials have coefficients in F_q and the averages are taken over finite-dimensional vector spaces where the standard nilsequence theory applies directly. In the revised version we will insert a new subsection (approximately 2.4) that spells out these verifications step by step, including a short discussion of why characteristic-p phenomena do not affect the nilfactor identification in this setting. This addition will make the transfer fully transparent while leaving the statements of Theorems 1.3 and 1.5 unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation appeals to established ergodic theory

full rationale

The paper's approach transfers combinatorial statements about Furstenberg-Katznelson constants to ergodic-theoretic statements on liminf of IP averages in finite-field vector spaces, studying characteristic factors and limits of multiple ergodic averages along IPs. No equations or steps in the abstract or described method reduce the claimed constants c, c_IP, c^rec or c_IP^rec to fitted parameters, self-definitions, or self-citation chains by construction. The work cites prior results by Bergelson et al. and Furstenberg-Katznelson (external to the author) and provides independent qualitative existence plus quantitative bounds in special cases such as Roth and IP-Roth, without renaming known patterns or smuggling ansatzes via self-citation. The derivation remains self-contained against external benchmarks in ergodic theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the transfer of ergodic-theoretic characteristic factors to finite-field vector spaces; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Characteristic factors exist for multiple ergodic averages along IP sequences in finite-field vector spaces
Invoked to obtain the limit behavior that yields the constants.

pith-pipeline@v0.9.0 · 5616 in / 1176 out tokens · 43936 ms · 2026-05-10T19:09:43.888793+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
We study the characteristic factors and limit of multiple ergodic averages along IPs in vector spaces over finite fields... Theorem 3.4: c_rec^IP(k,δ,F_ω^p) = c_IP(k,δ,F_ω^p) = ... = conv d_HPk(δ)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Definition 6.3 (IP Host–Kra seminorms) and Definition 6.4 (IP cubic measures) built inductively over the (γn)-junta factor

Reference graph

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