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Counterexamples to integer-coefficient criteria for recurrence along functions from a Hardy field

This paper provides negative answers to two questions posed by Bergelson, Moreira, and Richter by constructing counterexamples using elementary Bohr sets that demonstrate functions from a Hardy field can satisfy integer-coefficient derivative-span conditions yet fail to guarantee thick or even non-empty common return-time sets.

Original authors: Kangbo Ouyang, Leiye Xu, Shuhao Zhang

Published 2026-05-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kangbo Ouyang, Leiye Xu, Shuhao Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Predicting the Future of Numbers

Imagine you are a detective trying to predict when a specific pattern will reappear in a long, chaotic sequence of numbers. In mathematics, this is called recurrence.

The paper you are reading is about a specific type of number sequence generated by "Hardy field" functions. Think of these functions as machines that spit out numbers that grow at a steady, predictable rate (like $t^{3/2}$ , which grows faster than a square but slower than a cube).

For a long time, mathematicians believed they had a perfect "rulebook" (a set of criteria) to guarantee that these patterns would reappear often and in long, unbroken chains. This rulebook relied on looking at the real-number coefficients (the exact, messy decimals) of the functions.

The Question: The authors of this paper asked: Can we simplify this rulebook? Can we ignore the messy decimals and just look at the whole numbers (integers) inside the functions? If the whole-number rules are satisfied, does the pattern still have to reappear?

The Answer: No. The authors proved that the "whole-number" rulebook is not strong enough. You can satisfy all the integer rules, and the pattern might still fail to reappear, or it might reappear in a very broken, sparse way.

The Main Characters: The "Growth Machines"

To prove their point, the authors built two specific "machines" (pairs of functions) that act like tricksters.

1. The "Almost-There" Machine (Theorem 1.5)

Imagine two runners, Runner A and Runner B.

Runner A runs at a speed of $t^{3/2}$ .
Runner B runs at a speed of $\lambda t^{3/2} + t$ (where $\lambda$ is a weird, non-repeating decimal like $\pi$ ).

The Trap:
If you look at the whole number parts of their speeds, they seem to follow a perfect rule. They pass the "integer check."

The Expectation: You would expect that if you pick a starting point, you will eventually find a long stretch of time where both runners hit specific checkpoints simultaneously.
The Reality: The authors found a specific "track" (a set of numbers) where the runners do hit the checkpoints, but they never do so in a long, unbroken line. They hit them in a scattered, broken way.
The Metaphor: It's like a train schedule that looks perfect on paper (integer times), but when you actually try to catch the train, you only catch it for a split second, then have to wait a long time, then catch it again for a split second. You never get a long, continuous ride.

2. The "Ghost" Machine (Theorem 1.6)

This is an even more extreme version. The authors tweaked the second runner slightly (adding a shift $\xi$ ).

The Trap: This pair also passes the "integer check."
The Reality: On a specific track, the two runners never meet at the same checkpoint at the same time. The set of times they meet is empty.
The Metaphor: It's like two people trying to shake hands. They are both following the rules of the handshake protocol (the integer rules), but because of a tiny, invisible offset (the decimal part), their hands are always just a millimeter apart. They never actually touch.

3. The "Three-Way" Trick (Theorem 1.7)

The authors added a third runner to the mix to answer a different question: What if we look at the "shadow" of the functions (the polynomials they look like)?

They showed that even if the "shadows" of the functions look like they should work together perfectly (jointly intersective), the actual functions can still refuse to meet.
The Metaphor: Imagine three dancers. Their shadows on the wall look like they are perfectly synchronized. But in reality, on the dance floor, they are completely out of step and never meet in the center.

How They Did It: The "Bohr Obstruction"

How did they prove these runners never meet? They used a concept called a Bohr set.

Think of a Bohr set as a "safe zone" or a "fence" in a circular world (like a clock face).

The authors constructed a specific "fence" (a set of numbers $E$ ) that the runners are forced to stay near.
Because of the way the numbers are built (using that weird irrational number $\lambda$ ), the "fence" is shaped in a way that forces the runners to miss each other.
It's like setting up a maze where the walls are invisible. The runners follow the rules, but the geometry of the maze ensures they can never cross paths in the way the "integer rulebook" predicted they should.

The Takeaway

The paper delivers a "negative answer" to two big questions in the field of Ergodic Ramsey Theory (a branch of math that studies order in chaos).

You can't just use whole numbers. You cannot replace the complex, real-number rules with simple integer rules and expect the same guarantee of recurrence.
The "Shadow" isn't enough. Even if the polynomial "shadows" of the functions look promising, the actual functions might still fail to produce the expected patterns.

In short: The universe of these number sequences is more subtle than we thought. Just because the "whole number" parts of the rules look good doesn't mean the whole picture is good. The tiny, invisible decimal parts can completely break the pattern.

Technical Summary: Counterexamples to Integer-Coefficient Criteria for Recurrence Along Functions from a Hardy Field

Problem Statement
The paper addresses two specific questions posed by Bergelson, Moreira, and Richter regarding multiple recurrence along sequences generated by functions from a Hardy field. In ergodic Ramsey theory, recurrence results often rely on conditions involving the real linear span of functions and their derivatives. Specifically, Bergelson, Moreira, and Richter established that if a collection of functions $f_1, \dots, f_k$ from a Hardy field satisfies a condition where every non-zero element in their real derivative span diverges to infinity (or zero) at infinity, then the set of common return times for any set of positive upper density is "thick" (contains arbitrarily long intervals).

The authors investigate whether this strong hypothesis can be weakened to an integer-coefficient condition. They focus on two specific questions:

Question 1.2: Does the recurrence conclusion hold if one only assumes that every non-zero element in the integer derivative span ( $\nabla_{\mathbb{Z}}$ -span) diverges to infinity, rather than the real span?
Question 1.4: Does the recurrence conclusion hold if one replaces the assumption that the real linear span approximates a jointly intersective set of polynomials with the weaker assumption that the set of integer polynomials approximated by the real span ( $P_{\mathbb{Z}}$ ) is jointly intersective?

Methodology
The authors employ a constructive approach based on elementary Bohr sets to generate counterexamples. Their methodology involves:

Bohr Obstruction Principle: They utilize the fact that if a linear combination of integer-valued sequences $u_i(n)$ can be approximated by a real polynomial $P(n)$ modulo 1 (i.e., $\sum \theta_i u_i(n) - P(n)$ is bounded), then the set of common return times can be constrained to a set where $\|\beta P(n)\|_{\mathbb{T}}$ is small.
Equidistribution and Thickness: They leverage Weyl's equidistribution theorem and Boshernitzan's criterion for Hardy-field sequences to demonstrate that certain sets defined by polynomial growth (like $n^{3/2}$ ) are thick.
Construction of Counterexamples: By carefully selecting parameters (specifically irrational numbers $\lambda$ and integers $L$ ), they construct sets $E \subset \mathbb{N}$ of positive natural density such that the return-time sets are constrained by the "mod-one" obstruction. This obstruction prevents the return-time sets from being thick or even non-empty, despite the integer-coefficient conditions being satisfied.

Key Contributions and Results

Negative Answer to Question 1.2 (Theorem 1.5):
The authors provide a counterexample using the pair of functions $f_1(t) = t^{3/2}$ and $f_2(t) = \lambda t^{3/2} + t$ with $\lambda \in \mathbb{R} \setminus \mathbb{Q}$ .
- They prove that for every $F$ in the integer derivative span of these functions, $\lim_{t\to\infty} |F(t)| \in \{0, \infty\}$ .
- However, they construct a set $E$ of positive natural density such that the common return-time set $R_{f_1}(E) \cap R_{f_2}(E)$ is piecewise syndetic but not thick. This demonstrates that the integer-coefficient condition is insufficient to guarantee thickness, even in the specific test case singled out by Bergelson, Moreira, and Richter.
Negative Answer to Question 1.2 (Stronger Obstruction) (Theorem 1.6):
Using a closely related pair $g_1(t) = t^{3/2}$ and $g_2(t) = \lambda t^{3/2} + L(t + \xi)$ , the authors show that the integer-coefficient condition does not even guarantee that the common return-time set is non-empty.
- They construct a set $E$ of positive natural density where $R_{g_1}(E) \cap R_{g_2}(E) = \emptyset$ .
- This result relies on the fact that while the integer span condition holds, a real linear combination of the rounded iterates stays close to a polynomial expression modulo 1 that never hits zero, effectively ruling out common return times.
Negative Answer to Question 1.4 (Theorem 1.7):
The authors address the question of whether the joint intersectivity of integer polynomials ( $P_{\mathbb{Z}}$ ) implies the recurrence conclusion of Theorem 1.3.
- They construct a family $h_1(t) = t^{3/2}$ , $h_2(t) = \lambda t^{3/2} + L(t + \sqrt{2})$ , and $h_3(t) = t^2$ .
- They show that $P_{\mathbb{Z}}(h_1, h_2, h_3)$ consists of non-zero integer multiples of $t^2$ , which is jointly intersective.
- Despite this, they construct a set $E$ such that the intersection of the return-time sets for all three functions is empty. This proves that the weaker condition proposed in Question 1.4 does not imply the recurrence conclusion.

Significance
The paper provides definitive negative answers to two open questions regarding the sufficiency of integer-coefficient criteria in recurrence theory for Hardy-field sequences. The results demonstrate that:

The "integer-coefficient replacement" for the derivative-span hypothesis does not imply that common return-time sets are thick.
Under the same integer-coefficient hypothesis, the common return-time set may even be empty.
The joint intersectivity of integer polynomials approximated by the real span is insufficient to guarantee recurrence.

The authors conclude that the real-coefficient conditions in the existing theorems are necessary and cannot be relaxed to integer coefficients without losing the strength of the recurrence conclusions (thickness or even non-emptiness). The constructions rely on the interplay between the arithmetic properties of the functions (via Bohr sets) and the analytic properties of Hardy-field sequences.