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A criterion for weighted uniform distribution along functions from a Hardy field

This paper provides a new proof of Boshernitzan's theorem on the uniform distribution of Hardy field functions modulo 1 using summability theory, extends the result to weighted averages, and applies these findings to establish the existence of specific fractional parts for functions like $x^{3/2}$ within short intervals.

Original authors: Michael Reilly

Published 2026-06-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Michael Reilly

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

Imagine you have a giant, endless conveyor belt carrying numbers. You want to know if these numbers, when you look at just their "decimal parts" (what's left after you remove the whole numbers), are scattered randomly across the space between 0 and 1.

In mathematics, this is called being "uniformly distributed." If you pick a tiny slice of that space (say, between 0.2 and 0.3), and you wait long enough, you should find roughly the same proportion of numbers landing in that slice as you would in any other slice of the same size.

This paper, written by Michael Reilly, is about a specific type of number generator: functions that grow in a very smooth, predictable way (called Hardy fields). Think of these functions as machines that spit out numbers like $x^2$ , $x^{1.5}$ , or $\log(x)$ .

Here is the breakdown of what the paper does, using simple analogies:

1. The Old Rule vs. The New Rule

A mathematician named Michael Boshernitzan previously found a rule to tell us when these smooth machines produce random-looking decimal parts.

The Old Rule: It said, "If the machine's output grows fast enough and doesn't look too much like a simple polynomial (like $x^2$ or $3x+1$ ), then the decimals will be random."
The Problem: This rule was a bit rigid. It was like having a light switch that is either "On" (random) or "Off" (not random).

Reilly's Contribution:
Reilly introduces a "dimmer switch" for randomness. He creates a new way to measure randomness using weighted averages.

Imagine instead of counting every single number on the conveyor belt equally, you decide to pay more attention to the numbers at the very end of the belt, or perhaps you weigh them differently based on how fast the belt is moving.
Reilly proves that you can tune this "weight" to see exactly how random a function is. Some functions are "perfectly random" (uniformly distributed), some are "very random" (well-distributed), and some fall somewhere in between.

2. The Main Discovery (The "Golden Formula")

The paper provides a specific test to see if a function is random under these new, flexible rules.

Think of a function $f(x)$ as a car driving down a road.

The Derivative ( $f'$ ): This is the car's speed.
The Second Derivative ( $f''$ ): This is the car's acceleration.
The Weight ( $V$ ): This is a special ruler that measures time or distance in a non-standard way.

Reilly's formula says: To know if the car's position (mod 1) is random, you have to look at how the car's acceleration (or higher levels of change) compares to the rate of change of your special ruler.

The Condition: If the "jerkiness" or change in the car's motion is huge compared to the change in your ruler, the numbers will be random.
The Result: If this condition is met, the decimals of the numbers generated by the function will scatter perfectly across the 0-to-1 space, even if you are only looking at a specific, weighted chunk of the sequence.

3. A Concrete Example: The $x^{1.5}$ Machine

The paper uses a specific example to show off its power: the function $f(x) = x^{1.5}$ (which is $x$ times the square root of $x$ ).

The Question: If you take numbers like $1^{1.5}, 2^{1.5}, 3^{1.5}$ , etc., and look at their decimal parts, do they land in a specific range (like between 0.2 and 0.3) eventually?
The Old Way: It was hard to prove exactly how far back you needed to look to guarantee finding a number in that range.
Reilly's Way: The paper proves that for this specific machine, if you look at a chunk of numbers ending at a large number $N$ , and that chunk is roughly the size of $N^{0.25}$ (the fourth root of $N$ ), you are guaranteed to find a number with a decimal part in any range you choose.

The Analogy:
Imagine you are looking for a specific type of bird in a forest.

Uniform Distribution says: "If you walk far enough, you will eventually see the bird."
Reilly's Result says: "If you walk a specific distance (related to the size of the forest), you are guaranteed to see the bird right now."

4. Why This Matters (According to the Paper)

The paper doesn't claim this will cure diseases or build better computers. Instead, it solves a pure math puzzle:

It unifies two different ideas: It connects the idea of "uniform distribution" (randomness over the whole sequence) and "well distribution" (randomness over any chunk of the sequence) into one single framework.
It fixes a gap: It provides a complete proof for a theorem that was previously only half-proven in the literature.
It gives a precise map: It tells mathematicians exactly where any given "smooth" function sits on the spectrum of randomness.

Summary

Michael Reilly has built a new, more sensitive ruler to measure how "random" the decimal parts of smooth, growing numbers are. He proved that if a function changes fast enough relative to a specific weighting system, its numbers will be perfectly scattered. This allows us to make very precise predictions about where these numbers will land, even in short, specific segments of the sequence.

Technical Summary: A Criterion for Weighted Uniform Distribution along Functions from a Hardy Field

Problem Statement
The paper addresses the problem of characterizing the uniform distribution modulo 1 (u.d. mod 1) of sequences $(f(n))_{n \in \mathbb{N}}$ where $f$ is a function belonging to a Hardy field. While classical results by Boshernitzan provide criteria for standard uniform distribution and "well distribution" (w.d. mod 1) based on the growth rates of $f$ relative to polynomials, these results represent specific endpoints of a spectrum. The paper seeks to interpolate between these extremes by establishing necessary and sufficient conditions for $f$ to be uniformly distributed with respect to a broad class of weighted averages. Specifically, the authors aim to determine how the "quality" of uniform distribution depends on the growth rate of the weighting function $V$ relative to the asymptotic behavior of $f$ and its derivatives.

Methodology
The authors employ a synthesis of methods from summability theory, the theory of Hardy fields, and exponential sum estimates.

Hardy Field Framework: The analysis is restricted to functions $f$ and weight functions $V$ belonging to the same Hardy field, ensuring that limits of ratios of these functions exist. The paper utilizes the asymptotic comparison notation ( $\prec, \preceq$ ) to classify the growth of $f$ relative to polynomials $x^\ell$ .
Weighted Averages: The core methodology involves defining $V$ -weighted averages, $E^V_{n \le N} y_n$ , and uniform $V$ -weighted averages. The paper establishes a hierarchy of distribution properties using Lemma 1.7 and Lemma 1.9, showing that if a sequence is uniformly distributed with respect to a "faster" weight $V$ , it is also distributed with respect to a "slower" weight $U$ (where $\log U \preceq \log V$ ).
Inductive Proofs and Van der Corput's Trick: The proofs for the main theorem are conducted by induction on the integer $\ell$ $ℓ$ , which represents the smallest degree such that $|f(x) - q(x)| \preceq x^\ell$ $∣ f (x) - q (x) ∣ ⪯ x^{ℓ}$ for some rational polynomial $q$ $q$ .
- Case $\ell=1$ : The proof relies on Lemma 2.1, which approximates the weighted exponential sum by an integral involving the derivative $f'(x)$ and the weight $V$ . This reduces the problem to analyzing the limit of the ratio $|f'(x)| / (\log V(x))'$ .
- Case $\ell \ge 2$ : The proof of the converse direction (that failure of the criterion implies non-uniform distribution) utilizes a refined version of van der Corput's trick. The authors construct specific intervals where the higher-order differences $\Delta^i f(N)$ are small modulo 1 (Theorem 3.3). They then apply a Gaussian-weighted average (Corollary 3.6) and show that if the criterion fails, the weighted sum converges to a non-zero constant times a phase factor, preventing the limit from vanishing.
- Forward Direction: The proof that the criterion implies uniform distribution (Theorem 4.1) uses an inductive argument combined with the classical van der Corput inequality and bounds on exponential sums (Theorem 4.6) to show that the averages vanish when the derivative growth condition is met.

Key Contributions and Results

Theorem A (Main Result): The paper establishes a unified criterion for weighted uniform distribution. Let $V$ and $f$ be in the same Hardy field with $1 \prec \log(V(x)) \prec x$ . Let $\ell$ be the smallest integer such that $|f(x) - q(x)| \preceq x^\ell$ for some $q \in \mathbb{Q}[x]$ . The sequence $(f(n))$ is u.d. mod 1 with respect to $E^V$ if and only if:
$\lim_{x \to \infty} \frac{|f^{(\ell)}(x) - p(x)|^{1/\ell}}{(\log V(x))'} = \infty$
for all $p(x) \in \mathbb{Q}[x]$ .
Unification of Existing Theorems: The authors demonstrate that Theorem A generalizes Boshernitzan's classical results:
- Setting $V(x) = x$ recovers Boshernitzan's criterion for standard uniform distribution (Theorem 1.1).
- The framework also recovers conditions for well-distribution (Theorem 1.2) and results from recent literature (e.g., [BKS25], [Ric23]) as special cases.
Characterization of Distribution Quality: The paper provides a precise framework to interpolate between uniform distribution and well distribution. It shows that the "quality" of distribution is determined by the competition between the growth of the $\ell$ -th derivative of $f$ and the logarithmic derivative of the weight function $V$ .
Application to Specific Functions: As a concrete application, the paper analyzes $f(x) = x^{3/2}$ . It proves that for any interval $(a, b) \subseteq [0, 1]$ , there exists an $n$ in the range $[N - N^{1/4+\epsilon}, N]$ such that $f(n) \pmod 1 \in (a, b)$ for sufficiently large $N$ . The authors note this is an improvement on previous bounds and is optimal, as the result fails if the window size grows like $N^{1/4}$ or slower.

Significance and Claims
The paper claims to provide a complete characterization of Hardy functions regarding their uniform distribution properties under weighted averages. The significance lies in:

Generalization: It extends Boshernitzan's 1994 results from unweighted averages to a broad class of weighted averages, offering a continuous spectrum of distribution criteria.
Completeness: The authors provide a full proof for the equivalence in the well-distribution case (Theorem 1.2), noting that the forward implication was previously only cited without a published proof in the literature.
Methodological Synthesis: The work successfully integrates summability theory with the structural properties of Hardy fields to resolve distribution questions that were previously treated as distinct cases.

The authors maintain a modest tone, presenting the results as a theoretical extension of existing criteria rather than proposing new experimental applications or future directions beyond the immediate mathematical framework established.