REVIEW
Pointwise Ergodic Theorems Along $\lfloor p^c\rfloor$ for Every $c>1$
Classical Analysis and ODEs · Submitted 2 Nov 2026 · 29 references
Pith's verdict: The full-range c>1 theorem for prime-weighted fractional-power averages is real and well-built; the MSTT black box (Prop 2.7) needs a source-check and Prop 3.4's proof needs fleshing out, but nothing here looks fatal.
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The paper
The paper claims
The paper proves a full-range ergodic theorem along fractional powers of primes: for every noninteger $c>1$ and $1<p<\infty$, von Mangoldt-weighted averages converge a.e., and Möbius-weighted averages vanish at any logarithmic rate.
The reading
What carries the argument
The argument is carried by three linked mechanisms. First, short-interval polynomial approximation: on an interval of length $|\mathbf{I}|\asymp N^\theta$, $0<\theta<1$, the fractional phase $e(\xi n^c)$ is replaced, up to a power-saving error, by the polynomial phase $e(\xi P_k(n))$ with $P_k$ the Taylor polynomial of $t^c$ at the interval's center, so any correlation with a fractional phase reduces to a correlation with a polynomial phase (Lemma 4.1). Second, short-interval polynomial uniformity (Proposition 2.7, imported from [21]): on every interval $(X,X+H]$ with $X^{5/8+\varepsilon}\le H\le X^{1-\varepsilon}$, the weights $\Lambda-\Lambda_{\mathrm{Cramer},Q_J}$ and $\mu$ have polynomial-phase correlation at most $(\log X)^{-A}$ for every $A>0$, uniformly over all real polynomial coefficients; the fundamental lemma of sieve theory then converts the Cramér approximant $\Lambda_{\mathrm{Cramer},Q_J}$ — a pseudorandom model of the primes — into a divisor sum, and Lemma 3.3 evaluates that divisor sum's correlation with the fractional phase by an oscillatory integral. Third, the unweighted variation theorem for $B_N^c$ (Proposition 3.4, proved here from scratch) supplies the model dynamics that the $\Lambda$-weighted averages are forced to match; Proposition 4.2 makes the discrepancy at every dyadic scale smaller than $(\log N)^{-A}$, and the long-variation/short-variation decomposition turns that comparison into the claimed bound.
Extended reading notes
Core claim
The central claim, Theorem 1, asserts that for every noninteger $c>1$, every $1<p<\infty$ and every $r>2$, on every $\sigma$-finite measure-preserving system $(X,\mathcal{B},\nu,T)$ and every $f\in L^p(X)$, $$\bigl\|\mathcal{V}^r(A_N^c f:N\in\mathbb{N})\bigr\|_{L^p(X)}\lesssim_{c,p,r}\|f\|_{L^p(X)}.$$ Because the homogeneous $r$-variation seminorm controls the oscillation of the whole sequence of averages, the estimate yields almost-everywhere and $L^p$ convergence of $A_N^c f$, and Corollary 2 transfers the conclusion to the prime-normalized averages $P_N^c f(x)=\frac{1}{\pi(N)}\sum_{p\le N}f(T^{\lfloor p^c\rfloor}x)$, which share the same limit. Theorem 3 is the Möbius analogue: $\frac{(\log N)^A}{N}\sum_{n\le N}\mu(n)f(T^{\lfloor n^c\rfloor}x)\to 0$ almost everywhere for every $A>1$, every $1<p<\infty$ and every $f\in L^p(X)$. The load-bearing discovery behind all three statements is Proposition 4.2, $$\sup_{\xi\in\mathbb{T}}\left|\frac{1}{N}\sum_{n\le N}(\Lambda(n)-1)e(\xi\lfloor n^c\rfloor)\right|\lesssim_{c,A}(\log N)^{-A}\quad(A>0),$$ which says that at every frequency simultaneously the prime-weight discrepancy $\Lambda-1$ decorrelates from fractional-power phases with arbitrary logarithmic savings, forcing the weighted averages to track the unweighted averages $B_N^c f$, whose variation is controlled separately in Proposition 3.4.
Load-bearing premise
The load-bearing premise is the imported short-interval uniformity statement (Proposition 2.7, quoted from [21]): on every interval $(X,X+H]$ with $X^{5/8+\varepsilon}\le H\le X^{1-\varepsilon}$, the weights $\Lambda-\Lambda_{\mathrm{Cramer},Q_J}$ and $\mu$ have correlation at most $(\log X)^{-A}$ with every real-coefficient polynomial phase, uniformly in the interval's position and over all real coefficients, and the paper takes this in as a black box, asserting the polynomial-phase specialization without displaying the reduction from the published Gowers-norm form — if the cited result's actual length range or its uniformity over real coefficients were weaker than asserted, the discorrelation bound and with it both main theorems would fail.
Editorial extensions
If this is right
- For every noninteger $c>1$ and every $1<p<\infty$, the von Mangoldt-weighted averages $A_N^c f$ and the prime-normalized averages $P_N^c f$ converge almost everywhere and in $L^p$ on every $\sigma$-finite measure-preserving system, with the same limit for both normalizations.
- The Möbius-weighted averages $\frac{1}{N}\sum_{n\le N}\mu(n)f(T^{\lfloor n^c\rfloor}x)$ converge to $0$ almost everywhere for every $f\in L^p$, $1<p<\infty$, with a guaranteed rate $o((\log N)^{-A})$ for every fixed $A>1$.
- The previously known range for prime-weighted convergence along fractional powers is extended from $1<c<4/3$ to all noninteger $c>1$, and the proof avoids the circle-method minor-arc bottleneck that had limited the exponent.
- Because the variational estimate transfers to the integer shift and holds on every $\sigma$-finite system, the bounds are uniform in the system, covering infinite-measure dynamics as well as probability spaces.
- The unweighted variation theorem for $B_N^c$ receives a self-contained proof, so the convergence of the unweighted averages no longer depends on the earlier proof of that input.
Reading between the lines
- The proof's partition of each dyadic block into short subintervals of length at least $X^{5/8+1/100}$ indicates that the exponent $5/8$ in the imported uniformity estimate is the true bottleneck; if that estimate were later pushed to shorter intervals, the same short-interval replacement would plausibly extend to faster-growing sequences such as $\lfloor n^c(\log n)^a\rfloor$ or toward the unweigh
- Proposition 4.2 is itself a theorem about the distribution of primes: it says the primes are uniform against the curve $n\mapsto\lfloor n^c\rfloor$ with more-than-polylogarithmic precision, and that statement is likely reusable for equidistribution or recurrence questions involving several fractional powers at once, where current results mostly live in $L^2$.
- The paper notes that the second part of the original Möbius problem — the largest range of $c$ for which $L^1$ convergence holds — remains open; the present proof is firmly $L^p$-based, so reaching $f\in L^1$ would need weak-type input rather than a routine variation of this argument.
- Because the whole comparison runs through the Cramér model, the same discorrelation estimate should hold for any 'Cramér-like' weight (for example indicators of almost-primes), yielding a family of sparse ergodic theorems with essentially the same proof.
Original abstract
Let $c>1$ be noninteger, and let $\Lambda$ denote the von Mangoldt function. The averages \[ A_N^{c} f(x):=\frac1N\sum_{n=1}^{N} \Lambda(n)f\bigl(T^{\lfloor n^c\rfloor}x\bigr) \] satisfy the variational estimate \[ \bigl\|\mathcal{V}^r(A_N^{c} f:N\in\mathbb{N})\bigr\|_{L^p(X)} \lesssim_{c,p,r}\|f\|_{L^p(X)} \] for every $1<p<\infty$ and $r>2$, on every $\sigma$-finite measure-preserving system. In particular, the normalized averages along fractional powers of primes converge almost everywhere. The main arithmetic ingredient is a uniform bound on the correlations of $\Lambda-1$ with fractional-power exponential phases, yielding arbitrarily large powers of logarithmic gain. We obtain this estimate through short-interval polynomial approximation of the fractional phase and higher-order discorrelation estimates. The same approach yields almost-everywhere convergence to for the M\"obius-weighted averages along $\left( \lfloor n^c \rfloor \right)_{n \in \mathbb{N}}$ throughout $1<p<\infty$.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Let $c>1$ be noninteger, and let $\Lambda$ denote the von Mangoldt function. The averages \[ A_N^{c} f(x):=\frac1N\sum_{n=1}^{N} \Lambda(n)f\bigl(T^{\lfloor n^c\rfloor}x\bigr) \] satisfy the variational estimate \[ \bigl\|\mathcal{V}^r(A_N^{c} f:N\in\mathbb{N})\bigr\|_{L^p(X)} \lesssim_{c,p,r}\|f\|_{L^p(X)} \] for every $1<p<\infty$ and $r>2$, on every $\sigma$-finite measure-preserving system. In particular, the normalized averages along fractional powers of primes converge almost everywhere. The main arithmetic ingredient is a uniform bound on the correlations of $\Lambda-1$ with fractional-power exponential phases, yielding arbitrarily large powers of logarithmic gain. We obtain this estimate through short-interval polynomial approximation of the fractional phase and higher-order discorrelation estimates. The same approach yields almost-everywhere convergence to for the M\"obius-weighted averages along $\left( \lfloor n^c \rfloor \right)_{n \in \mathbb{N}}$ throughout $1<p<\infty$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $c>1$ be noninteger, and let $\Lambda$ denote the von Mangoldt function. | background 0.99 | 0.10 | 0.02 | unclear 0.59 |
| s1 | The averages \[ A_N^{c} f(x):=\frac1N\sum_{n=1}^{N} \Lambda(n)f\bigl(T^{\lfloor n^c\rfloor}x\bigr) \] satisfy the variational estimate \[ \bigl\|\mathcal{V}^r(A_N^{c} f:N\in\mathbb{N})\bigr\|_{L^p(X)} \lesssim_{c,p,r}\|f\|_{L^p(X)} \] for every $1<p<\infty$ and $r>2$, on every $\sigma$-finite measure-preserving system. | result 0.99 | 0.88 | 0.60 | asserted 1.00 |
| s2 | In particular, the normalized averages along fractional powers of primes converge almost everywhere. | interpretation 0.68 | 0.86 | 0.48 | asserted 1.00 |
| s3 | The main arithmetic ingredient is a uniform bound on the correlations of $\Lambda-1$ with fractional-power exponential phases, yielding arbitrarily large powers of logarithmic gain. | contribution 0.62 | 0.87 | 0.06 | asserted 1.00 |
| s4 | We obtain this estimate through short-interval polynomial approximation of the fractional phase and higher-order discorrelation estimates. | method 1.00 | 0.91 | 0.03 | asserted 1.00 |
| s5 | The same approach yields almost-everywhere convergence to for the M\"obius-weighted averages along $\left( \lfloor n^c \rfloor \right)_{n \in \mathbb{N}}$ throughout $1<p<\infty$. | result 0.94 | 0.92 | 0.14 | asserted 0.99 |
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