pith. sign in

arxiv: 2606.08040 · v1 · pith:D45SS55Unew · submitted 2026-06-06 · 🧮 math.NT · math.DS

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

Michael Reilly This is my paper

Pith reviewed 2026-06-27 19:38 UTC · model grok-4.3

classification 🧮 math.NT math.DS
keywords uniform distribution modulo 1Hardy fieldsweighted averagesBoshernitzan criterionsummability theorynumber theory
0
0 comments X

The pith

Boshernitzan's criterion extends to weighted uniform distribution for Hardy field functions via summability methods.

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

The paper supplies a new proof of the classical result that a function f from a Hardy field with |f(x)| growing slower than some power of x is uniformly distributed modulo 1 precisely when it diverges from every rational polynomial faster than logarithmically. It then derives necessary and sufficient conditions for the same conclusion to hold under a broad class of weighted averages instead of the usual Cesaro means. The extension is applied to show that the fractional parts of x to the three-halves become dense in any subinterval of the unit interval inside sufficiently short segments of the integers near N.

Core claim

If f belongs to a Hardy field and satisfies |f(x)| ≺ x^ℓ for some natural number ℓ, then f(n) is uniformly distributed modulo 1 with respect to a broad class of weighted averages if and only if lim |f(x) - p(x)| / log(x) = ∞ as x → ∞ for every p(x) in the rational polynomials.

What carries the argument

Summability theory applied to weighted averages to obtain the necessary and sufficient divergence condition.

Load-bearing premise

The function f belongs to a Hardy field and grows slower than some fixed power of x.

What would settle it

A concrete function in a Hardy field where |f(x) - p(x)| remains bounded by a multiple of log(x) for some rational polynomial p, yet the weighted averages of the sequence still converge to the uniform measure on the circle.

read the original abstract

A classical theorem of Boshernitzan states that if $f$ is a function which belongs to a Hardy field and which satisfies $|f(x)|\prec x^{\ell}$ for some $\ell\in \mathbb{N}$, then the sequence $(f(n))_{n\in \mathbb{N}}$ is uniformly distributed modulo 1 if and only if $\lim_{x\to\infty}\frac{|f(x)-p(x)|}{\log(x)} = \infty$ for all $p(x)\in \mathbb{Q}[x]$. We provide a new proof of this result using methods from summability theory and we extend Boshernitzan's criterion by obtaining necessary and sufficient conditions for $f$ to be uniformly distributed modulo 1 with respect to a broad class of weighted averages. As an application of our results, we show that for the function $f(x) = x^{3/2}$ and for any $(a,b)\subset [0,1]$, and all sufficiently large $N\in\mathbb{N}$, there is an $n\in [N-N^{\frac{1}{4}},N]$ such that $f(n)\mod 1\in (a,b)$.

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

2 major / 1 minor

Summary. The paper gives a new proof via summability theory of Boshernitzan's theorem that f(n) is uniformly distributed mod 1 for f in a Hardy field with |f(x)| ≺ x^ℓ iff |f(x)-p(x)|/log x → ∞ for every rational polynomial p. It extends the criterion to necessary and sufficient conditions for uniform distribution with respect to a broad class of weighted averages. As an application it asserts that f(x)=x^{3/2} satisfies {f(n)} ∈ (a,b) for some n in every interval [N-N^{1/4},N] and all large N.

Significance. If the weighted extension is correctly proved, the result strengthens the classical criterion by replacing ordinary averages with a flexible family of weights while retaining the Hardy-field setting. The short-interval application, if the missing local-control argument is supplied, would give a concrete density statement beyond the global average.

major comments (2)
  1. [Application to f(x)=x^{3/2}] Application paragraph (following the weighted criterion): the claim that global weighted uniform distribution implies existence of n ∈ [N-N^{1/4},N] with {f(n)} ∈ (a,b) does not follow automatically from the weighted averages; an auxiliary estimate relating the weight decay, the Hardy-field growth bound, and the oscillation of f over intervals of length N^{1/4} is required. No such lemma or explicit reduction is indicated in the abstract and must be supplied if the application is to be retained.
  2. [Main theorem on weighted averages] Statement of the weighted criterion (presumably §3 or §4): the precise class of admissible weights and the exact form of the necessary-and-sufficient condition (e.g., whether it involves a weighted analogue of the lim |f-p|/log x = ∞) must be stated with the same precision as Boshernitzan's original condition so that the extension can be verified against the summability argument.
minor comments (1)
  1. Notation for the weighted averages should be introduced once and used consistently; the abstract refers to “a broad class of weighted averages” without a symbol or defining property.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive suggestions. We address the two major comments point by point below. Both points identify places where additional detail or an auxiliary argument will strengthen the manuscript, and we will incorporate the necessary revisions.

read point-by-point responses

  1. Referee: [Application to f(x)=x^{3/2}] Application paragraph (following the weighted criterion): the claim that global weighted uniform distribution implies existence of n ∈ [N-N^{1/4},N] with {f(n)} ∈ (a,b) does not follow automatically from the weighted averages; an auxiliary estimate relating the weight decay, the Hardy-field growth bound, and the oscillation of f over intervals of length N^{1/4} is required. No such lemma or explicit reduction is indicated in the abstract and must be supplied if the application is to be retained.

    Authors: We agree that the passage from the global weighted uniform distribution result to the short-interval density statement for f(x)=x^{3/2} requires an auxiliary estimate that controls the oscillation of f on intervals of length N^{1/4} in terms of the weight decay and the Hardy-field growth. In the revised manuscript we will insert a new lemma establishing this local control, using the fact that f'(x) remains bounded on those intervals together with the regularity properties of the weights. This will make the application rigorous while preserving the stated interval length N^{1/4}. revision: yes

  2. Referee: [Main theorem on weighted averages] Statement of the weighted criterion (presumably §3 or §4): the precise class of admissible weights and the exact form of the necessary-and-sufficient condition (e.g., whether it involves a weighted analogue of the lim |f-p|/log x = ∞) must be stated with the same precision as Boshernitzan's original condition so that the extension can be verified against the summability argument.

    Authors: The weighted criterion appears in Section 3. The admissible weights are those positive sequences (w_n) for which the associated summability method is regular, the row sums are 1, and the weights satisfy a mild growth restriction compatible with Hardy-field functions. The necessary-and-sufficient condition is the weighted analogue lim |f(x)-p(x)|/log x = ∞ (with the same logarithmic denominator). We will revise the statement to display the precise hypotheses on the weights in a numbered list matching the format of Boshernitzan's original condition and will add a short paragraph comparing the two statements directly. revision: yes

Circularity Check

0 steps flagged

No circularity: new proof of external classical result via summability methods

full rationale

The paper states it supplies a new proof of Boshernitzan's theorem (an external classical result) using summability theory, then extends the criterion to weighted averages by deriving necessary and sufficient conditions. The growth restriction |f(x)| ≺ x^ℓ is an explicit hypothesis taken from the classical statement, not fitted or redefined inside the paper. The application to short-interval existence for f(x)=x^{3/2} is presented as a consequence of the extended criterion; no equation reduces a claimed prediction to a fitted parameter by construction, and no self-citation chain is invoked as load-bearing justification. The derivation chain is therefore self-contained against the external benchmark of Boshernitzan's theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review prevents exhaustive extraction; the central claim rests on the definition of Hardy fields and the growth bound |f(x)| ≺ x^ℓ, both standard in the cited literature.

axioms (2)
  • domain assumption f belongs to a Hardy field
    Invoked in the statement of Boshernitzan's theorem and its extension.
  • domain assumption |f(x)| ≺ x^ℓ for some natural ℓ
    Growth restriction stated explicitly in the abstract as the setting for the criterion.

pith-pipeline@v0.9.1-grok · 5727 in / 1277 out tokens · 15643 ms · 2026-06-27T19:38:55.050879+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

15 extracted references · 2 canonical work pages

  1. [1]

    Bergelson and J

    V. Bergelson and J. Moreira and F. Richter , title =. Adv. Math. , volume =. 2020 , pages =

    2020

  2. [2]

    Boos and P

    J. Boos and P. Cass , title =

  3. [3]

    Boshernitzan , title =

    M. Boshernitzan , title =. J. Anal. Math. , volume =. 1994 , pages =

    1994

  4. [4]

    Boshernitzan , title =

    M. Boshernitzan , title =

  5. [5]

    Frantzikinakis , title =

    N. Frantzikinakis , title =. J. Anal. Math. , volume =. 2022 , pages =

    2022

  6. [6]

    Richter , year =

    F. Richter , year =. Uniform distribution in nilmanifolds along functions from a. J. Anal. Math. , pages =

  7. [7]

    Schatte , title =

    P. Schatte , title =. Math. Nachr. , volume =. 1974 , pages =

    1974

  8. [8]

    Schatte , title =

    P. Schatte , title =. Z. Anal. Anwendungen , volume =. 1990 , pages =

    1990

  9. [9]

    Kuipers and H

    L. Kuipers and H. Niederreiter , year =. Uniform Distribution of Sequences , publisher =

  10. [10]

    Uniform Weighted Averages and a Conjecture of

    Michael Reilly , year=. Uniform Weighted Averages and a Conjecture of. 2602.20606 , archivePrefix=

    work page arXiv

  11. [11]

    Boshernitzan , title =

    M. Boshernitzan , title =. J. Lond. Math. Soc. , year =

  12. [12]

    Bergelson and G

    V. Bergelson and G. Kolesnik and Y. Son , title =. J. Anal. Math , pages =. 2019 , doi =

    2019

  13. [13]

    Graham and G

    S. Graham and G. Kolesnik , year=. Van der

  14. [14]

    Bergelson, Vitaly and Moreira, Joel , journal =. Van der. 2016 , issn =. doi:10.1016/j.indag.2015.10.014 , file =

    work page doi:10.1016/j.indag.2015.10.014 2016

  15. [15]

    2025 , eprint=

    Weighted uniform distribution of subpolynomial functions along primes and applications , author=. 2025 , eprint=

    2025