{"id":"5fd8a93b-a32e-4307-8203-9530217e2283","arxiv_id":"2611.02465","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Prime-weighted averages (1/N) Σ_{n≤N} Λ(n) f(T^{⌊n^c⌋}x) converge almost everywhere for every noninteger c > 1 and every f ∈ L^p with p > 1, with a quantitative variational estimate.","lead":"This paper proves that ergodic averages taken along fractional powers of primes converge almost everywhere for every exponent c > 1 and every L^p function with p > 1. The result closes a line of work that had stalled below exponent 4/3, and the discorrelation technique behind it may transfer to other sparse averaging problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"The load-bearing point is Proposition 2.7: both main theorems need [21] to give a pointwise, every-interval u_s-uniformity bound in the range X^{5/8+ε} ≤ H ≤ X^{1−ε}; if the cited theorem is only an X-average, Proposition 4.2 and Theorem 1 would not follow.","rationale":"Reader's weakest assumption is the same one I identify. I read Proposition 4.2 in detail. Conditional on Proposition 2.7, the short-interval partition, Lemma 3.3, Lemma 4.1, and the sieve step are coherent: floor-removal errors are controlled by Lemma 3.2, the Cramér-to-divisor-sum switch uses Lemma 2.5 with admissible parameters, and the final exponent condition γδ_c + θ² > 1 is satisfiable. The closing display of Proposition 4.2 contains an exponent that does not match the preceding line (θ/2 versus θ²); this is fixable bookkeeping, not a structural failure. Lemma 3.2 and Proposition 3.4 are sketched, but the essential inputs are cited published tools ([23, Theorem 2.39], [26, Corollary 1.6]) and the proofs given are enough to locate the assumptions. The genuine single point on which both Theorem 1 and Theorem 3 depend is the scope and uniformity of [21] as imported by Proposition 2.7. Because [21] is published, the correct test is a source verification, not a re-proof. If the source does contain the every-interval, real-coefficient polynomial-phase statement with the stated H-range, the central argument appears sound and the paper should be accepted after the copy edits and the Proposition 4.2 exponent correction. I therefore keep the reader's CONDITIONAL verdict unchanged.","tokens_in":19833,"tokens_out":28941,"duration_ms":289527,"concrete_test":"Compare Proposition 2.7 verbatim with [21, Theorem 1.1(i)-(ii)]. (i) Confirm that the source bound is uniform over every interval J=(X,X+H], not merely an average over X in [K,2K] or an almost-all statement. (ii) Confirm that it controls sup over real polynomials P of degree at most s−1 of |E_{n∈J} w(n) e(−P(n))| for w = Λ−Λ_Cramér,Q_J and w = μ, or that this follows from the stated Gowers U^s norm with the same H-range X^{5/8+ε} ≤ H ≤ X^{1−ε}. (iii) Confirm that Q_J = exp((log X)^{1/10}) is within the admissible sieve cutoff class. If (i) holds only on average, identify the exceptional-set bound; no averaged version sustains Proposition 4.2 as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2.7 imports [21, Theorem 1.1(i)-(ii)] as a black box, and the paper describes it as a 'polynomial-phase specialization'. The proposition asserts the strongest possible form: for every interval J=(X,X+H] with X^{5/8+ε} ≤ H ≤ X^{1−ε}, every fixed s and every A>0, the bound ‖Λ−Λ_Cramér,Q_J‖_{u_s(J)} + ‖μ‖_{u_s(J)} ≪ (log X)^{−A} holds uniformly in X, H and all real-coefficient polynomial phases. This is exactly what Proposition 4.2 needs for each subinterval in the partition of [K,K′]: those subintervals occur at arbitrary positions, so every one must satisfy the bound individually. An X-averaged or almost-all version of the source theorem would not suffice. Section 6 has the same dependence through the Möbius case. The reduction from the published Gowers-norm statements is not displayed, and the paper does not record whether the source's uniformity is over every interval position or only on average over X, nor whether the u_s norm with real coefficients follows from the source's stated norms without extra hypotheses. If the source's actual range or uniformity is weaker than asserted, the discorrelation bound of Proposition 4.2 and both main theorems fail. Since [21] is published, the resolution is a check against the source, not a request for a new proof; if the source contains this uniform statement, the concern disappears.","agreement_with_reader":"agree"},"referee_report":null,"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Quick take: the headline claim is real. The paper proves the r-variation estimate (r > 2, 1 < p < ∞) for von Mangoldt-weighted averages along floor(n^c) for every noninteger c > 1 on any sigma-finite system, and answers Teravainen's Mobius question with an explicit (log N)^{-A} rate. That settles the fractional-prime range question in its strongest known form, well past [1]'s 1 < c < 4/3.\n\nThe genuinely new piece is Proposition 4.2: sup over xi of |E_{n<=N} (Lambda(n)-1) e(xi floor(n^c))| is O((log N)^{-A}) for every A, for every noninteger c > 1. Trading the circle method for short-interval u_s uniformity plus the sieve lemma is a different strategy, and it is what buys the full range. The paper is also honest about what stays open, notably the L^1 half of Teravainen's question.\n\nThree soft spots, in proportion.\n\nFirst, Proposition 2.7 is the load-bearing import. It asserts a 'polynomial-phase specialization' of [21, Theorem 1.1(i)-(ii)] with pointwise, every-interval uniformity in X^{5/8+epsilon} <= H <= X^{1-epsilon}, and both main theorems route through it. The reduction from the published statement is not displayed. I do not think the stress-test's worry lands: the source is the 'All intervals' paper, and the asserted range and the (Lambda - Cramer, mu) split match that paper's headline theorem. This is a check against the source, not an evident flaw, but it is the one item I would insist a referee verify.\n\nSecond, the closing arithmetic of Proposition 4.2 is compressed: the condition gamma*delta_c + theta^2 > 1 is satisfiable, and any reasonable bookkeeping still yields the claimed (log N)^{-A}, so this is fixable presentation, not a broken step.\n\nThird, Proposition 3.4 (the unweighted variation estimate that anchors the comparison) is proved by a sketch relying on [23, Thm 2.39], [17] and [20], while the cited source for the result, [26], is a preprint. A referee should ask for a fuller argument or an updated citation.\n\nMinor: the abstract has 'convergence to for', and the title slightly overclaims since the theorem covers noninteger c > 1, with integer powers already in [24], [25].\n\nWho it is for: ergodic theorists working on sparse pointwise theorems, and analytic number theorists interested in short-interval uniformity as a tool. It deserves a serious referee. Send it out; ask the referee specifically to check Proposition 2.7 against [21] and to read Proposition 3.4 carefully.","headline":"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.","tokens_in":20936,"tokens_out":10010,"would_cite":true,"duration_ms":57286,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["37A30","37A45","11L20","11N05"],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"The paper proves a full-range ergodic theorem along fractional powers of primes: for every noninteger $c>1$ and $10$; observing instead a decay like $(\\log N)^{-B}$ for a finite exponent $B$, or no decay across large scales, would refute the engine of the paper and both theorems. A second check targets the black-box input directly: if any interval family with $H$ slightly below $X^{5/8}$ exhibited correlation of $\\Lambda-\\Lambda_{\\mathrm{Cramer},Q_J}$ with a linear phase at strength $(\\log X)^{-A}$, the subinterval partition in the proof of Proposition 4.2, which insists on subinterval lengths $\\ge X^{5/8+1/100}$, could not be carried out.","tokens_in":21094,"feed_emoji":"🔄","tokens_out":22925,"duration_ms":108488,"temperature":0.7,"pith_summary":"This paper proves that ergodic averages sampled along fractional powers of primes converge for every noninteger exponent $c>1$, the full range beyond the previously settled $12$, and consequently the averages converge almost everywhere and in $L^p$, with the prime-normalized averages converging to the same limit. The engine is a uniform discorrelation bound showing that $\\Lambda(n)$ — the von Mangoldt weight that assigns $\\log p$ to prime powers — is indistinguishable from the constant weight $1$ against the phases $e(\\xi\\lfloor n^c\\rfloor)$, up to errors smaller than any power of $\\log N$, uniformly in the frequency $\\xi$. The same mechanism, with the Möbius function $\\mu(n)$ in place of $\\Lambda(n)-1$, makes the Möbius-weighted averages along $\\lfloor n^c\\rfloor$ converge to zero almost everywhere at rate $(\\log N)^{-A}$ for every $A>1$, settling an open question from a conference problem list. A sympathetic reader should care because the result fuses the two classical sources of sparsity — primes and non-integer powers — into one quantitative theorem valid throughout the reflexive range $11","feed_subtitle":"Von Mangoldt weights now match unweighted averages with arbitrary log decay, for all noninteger c>1.","key_machinery":"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.","core_discovery":"The central claim, Theorem 1, asserts that for every noninteger $c>1$, every $12$, on every $\\sigma$-finite measure-preserving system $(X,\\mathcal{B},\\nu,T)$ and every $f\\in L^p(X)$,\n$$\\bigl\\|\\mathcal{V}^r(A_N^c f:N\\in\\mathbb{N})\\bigr\\|_{L^p(X)}\\lesssim_{c,p,r}\\|f\\|_{L^p(X)}.$$\nBecause 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 $10),$$\nwhich 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.","pith_inferences":["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."],"forward_implications":["For every noninteger $c>1$ and every $11$.","The previously known range for prime-weighted convergence along fractional powers is extended from $11$, 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."],"supporting_citations":[{"why":"Supplies the short-interval polynomial uniformity quoted as Proposition 2.7: $\\Lambda-\\Lambda_{\\mathrm{Cramer},Q_J}$ and $\\mu$ decay like $(\\log X)^{-A}$ against polynomial phases on intervals of length $\\ge X^{5/8+\\varepsilon}$; both main theorems depend on it.","marker":"[21]"},{"why":"The previous pointwise convergence theorem for $11: prime-powered ergodic averages converge a.e.","All c>1: ergodic averages along floor(p^c) converge a.e.","Prime-indexed fractional powers: ergodic convergence for all c>1","Von Mangoldt-weighted ergodic averages converge for every c>1","Ergodic theorem for fractional prime powers: every c>1"],"cache_read_input_tokens":0,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Every c>1: prime-powered ergodic averages converge a.e.","All c>1: ergodic averages along floor(p^c) converge a.e.","Prime-indexed fractional powers: ergodic convergence for all c>1","Von Mangoldt-weighted ergodic averages converge for every c>1","Ergodic theorem for fractional prime powers: every c>1"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.001804,"raw_usage":{"cost":0.00180404,"is_byok":false,"cost_details":{"upstream_inference_cost":0.00180404,"upstream_inference_prompt_cost":0.0001666,"upstream_inference_completions_cost":0.00163744},"total_tokens":7038,"prompt_tokens":1190,"completion_tokens":5848,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":0,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":5750}},"tokens_in":1190,"tokens_out":5848,"cost_source":"openrouter_usage","duration_ms":23958,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":5750,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-11-03T21:59:42.340320+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"A decisive check is computational: fix a noninteger $c$ (say $c=3/2$) and a frequency inside the torus (say $\\xi=0.3$), and evaluate $S(N)=\\frac{1}{N}\\sum_{n\\le N}(\\Lambda(n)-1)e(\\xi\\lfloor n^c\\rfloor)$ up to large $N$. Proposition 4.2 predicts $|S(N)|\\le C_A(\\log N)^{-A}$ for every $A>0$; observing instead a decay like $(\\log N)^{-B}$ for a finite exponent $B$, or no decay across large scales, would refute the engine of the paper and both theorems. A second check targets the black-box input directly: if any interval family with $H$ slightly below $X^{5/8}$ exhibited correlation of $\\Lambda-\\Lambda_{\\mathrm{Cramer},Q_J}$ with a linear phase at strength $(\\log X)^{-A}$, the subinterval partition in the proof of Proposition 4.2, which insists on subinterval lengths $\\ge X^{5/8+1/100}$, could not be carried out.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the short-interval polynomial uniformity quoted as Proposition 2.7: $\\Lambda-\\Lambda_{\\mathrm{Cramer},Q_J}$ and $\\mu$ decay like $(\\log X)^{-A}$ against polynomial phases on intervals of length $\\ge X^{5/8+\\varepsilon}$; both main theorems depend on it."},{"cited_title":"Pointwise ergodic theorems along fractional powers of primes","cited_arxiv_id":"2412.07055","evidence_quote":"The previous pointwise convergence theorem for $1