{"id":"5b6de073-5eda-4ef6-96e3-a66518f4eb0e","arxiv_id":"2504.18192","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves, in self-contained form, that any non-atomic self-similar measure whose Fourier transform vanishes at infinity is pointwise absolutely normal.","lead":"This paper surveys recent results showing that typical points of self-similar fractal measures are normal numbers unless the fractal's structure forces particular digits. It includes a full proof that a self-similar measure with vanishing Fourier transform is normal in every integer base.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The reduction to [0,1] by affine conjugacy is asserted without showing that absolute normality transfers; a correct reduction needs an integer-affine map.","rationale":"The main proof of Theorem 1.2 is sound: Lemma 3.1's Fourier criterion is valid, Lemma 3.2's application of Hochman's Theorem 2.1 satisfies condition (2.2) via the uniform growth τ_n ≥ c n and the diameter estimate O(b^{-k}), and the final Rajchman estimate is correct. The reader's stated weakest assumption, the partition condition in Lemma 3.2, is actually satisfied for this finite IFS. The only genuine soft spot is the terse affine normalization to [0,1]: as written it is not justified, and a naive arbitrary affine conjugacy would not preserve absolute normality. However, the proof is coordinate-free in substance and can be repaired either by removing the normalization or by making the integer-affine reduction explicit. Thus the reader's CONDITIONAL verdict remains appropriate, with the requested clarification centered on the normalization step and the minor typos in Section 3.3.","tokens_in":19556,"tokens_out":40611,"duration_ms":429832,"concrete_test":"Rewrite Section 3 with an arbitrary invariant compact interval I and no normalization; verify that the estimates in Lemma 3.2 (diam(T^n C) ≤ max{ρ^{c(n+k)}, C·diam(I)·b^{-k}}) and the Fourier identity (3.8) go through unchanged. If they do, delete the WLOG sentence or replace it with the explicit integer-affine reduction L(x)=(x−a)/(b−a), a,b∈Z, checking that the conjugated measure inherits both the Rajchman property and the normality conclusion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3 opens with 'Without the loss of generality we may assume that Φ preserves the interval I=[0,1]'. If this is done by an arbitrary affine conjugacy L, the implication fails: from L_∗ν being pointwise absolutely normal one cannot conclude ν is pointwise absolutely normal, since absolute normality is not invariant under arbitrary affine maps (e.g., dividing by 2 can destroy normality to odd bases). The intended reduction must use an affine map whose inverse is multiplication by an integer plus an integer translation, e.g., L(x)=(x−a)/(b−a) with a,b∈Z; such a map does preserve absolute normality. The paper does not specify this structure, so the WLOG step is unjustified as written. The subsequent proof is in fact coordinate-free and works for an arbitrary invariant compact interval, so the gap is removable, but it must be fixed for the proof to be rigorous as presented.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper is an expository article on recent progress toward proving pointwise normality of typical points in the support of self-similar measures on the real line. Its main technical contribution is an essentially self-contained proof of Theorem 1.2: every non-atomic self-similar measure whose Fourier transform vanishes at infinity (a Rajchman measure) is pointwise absolutely normal. The proof combines Hochman's equidistribution theorem (Theorem 2.1) with a stopping-time transfer argument that reduces the problem to the Rajchman property at rescaled frequencies. The paper also explains how Theorem 1.2, together with a normality criterion from Algom--Baker--Shmerkin and Barany--Kaenmaki--Pyorala--Wu, and the recent classification of Rajchman self-similar measures by Li--Sahlsten and Bremont, yields a structural description of self-similar measures that fail to be pointwise absolutely normal (Theorem 1.1). Several open problems are discussed, concerning effective equidistribution, non-integer bases, and higher-order correlations.","tokens_in":19631,"tokens_out":31671,"duration_ms":280073,"significance":"If the proof is correct after the gaps noted below are fixed, Theorem 1.2 is a substantial improvement over the classical Davenport--Erdos--LeVeque criterion for self-similar measures, because it requires no quantitative decay rate on the Fourier transform. The paper gives a valuable, largely self-contained exposition of Hochman's method and makes explicit how the Rajchman property alone forces absolute normality for this class. The discussion of open problems, especially the non-integer base case and higher-order correlations, is thoughtful and well connected to recent literature. The paper also includes a complete proof of Theorem 2.1, which is a useful pedagogical contribution. These strengths make the paper potentially valuable to researchers in metric number theory and fractal geometry.","major_comments":[{"comment":"The reduction \"Without the loss of generality we may assume that Φ preserves the interval I=[0,1]\" is not justified by an arbitrary affine conjugacy, because pointwise absolute normality is not invariant under general real affine maps. The subsequent proof is coordinate-free and works for any invariant compact interval; please either carry a general interval I through the proof or explicitly state that the proof is written for I=[0,1] only for notational convenience and that no transfer of normality under a change of variables is being used. As written, the proof of Theorem 1.2 covers only measures with an invariant interval [0,1].","section":"Section 3, opening paragraph"},{"comment":"The sufficient condition (3.7) bounds only the single term |F_E(T_b^B ∘ φ_{ω|τ_B(ω)}_* ν)|, but Lemma 3.2 involves the Cesàro average (1/N)Σ_{n=0}^{N-1} T_b^n ∘ φ_{ω|τ_B(ω)}_* ν. To conclude the desired limsup bound, one must additionally argue that for n≥B the frequencies E b^n φ'_{ω|τ_B(ω)}(0) have magnitude at least |E|c0 so the same Rajchman estimate applies, and that the finitely many n<B terms contribute negligibly to the Cesàro average. This argument is missing. Relatedly, the statement of Lemma 3.2 should quantify B explicitly (\"for every B∈N\").","section":"Section 3.3, Eq. (3.7)"}],"minor_comments":[{"comment":"The sentence \"We are now in position to prove Theorem 1.3\" should read \"Theorem 1.2\".","section":"Section 3.3, first sentence"},{"comment":"The phrase \"if R=R·c0\" appears to be a typo; it should be \"if R=R'/c0\".","section":"Section 3.3, after Eq. (3.7)"},{"comment":"The lemma statement uses τ_B(ω) but does not quantify B; add \"for every B∈N\" to the statement.","section":"Lemma 3.2, statement"},{"comment":"The text says \"the diameter of the projection of T^B A to the first coordinate\" twice; the second occurrence should refer to the second coordinate.","section":"Lemma 3.2, proof"},{"comment":"The phrase \"We note, however, the such bounds are usually hard to obtain\" contains a grammatical error; \"the such\" should be \"such\".","section":"Section 1, paragraph after Theorem 1.2"},{"comment":"There is a typographical spacing issue in \"The orem\" in the abstract; this should be corrected.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The paper is an exposition of recent results, and the main theorem (Theorem 1.2) is already proved in the author's earlier work [8] with collaborators. The novel contribution here is the self-contained proof and the clarity of the exposition. The editor may wish to judge whether the journal's scope includes expository survey articles; if so, this paper is a useful contribution. The gaps identified in the major comments are fixable within the scope of the manuscript and do not undermine the central ideas."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is an exposition of a recent result (Theorem 1.2) that is already published in [8], and the paper itself says so. The main value is the self-contained proof and the clean survey of the surrounding machinery. No new theorems are proved, but that's fine for an exposition if the proof is reliable. On the whole it is, with one important caveat.\n\nWhat it does well: the proof of Theorem 1.2 is assembled carefully. Lemma 3.1 is a Fourier criterion for pointwise normality that is transparent. Lemma 3.2 applies Hochman's martingale theorem to the graph measure on A^N × T, using a stopping time to verify the diameter condition without any separation assumption. The final Rajchman estimate at rescaled frequencies is correct. The survey of Theorem 1.1 and Theorem 1.3, and the open problems (effective rates, non-integer bases, higher correlations), are informative and honest about what is known. The references are cited properly; the self-citation to [8] is for provenance, not to smuggle in the result.\n\nThe soft spots. The most substantial is the WLOG step at the start of Section 3: 'without loss of generality we may assume Φ preserves I=[0,1]'. This is not a harmless normalization. The theorem's conclusion is absolute normality of typical points, and absolute normality is not invariant under arbitrary real affine maps. The map that sends the invariant interval to [0,1] generally involves an irrational scaling and translation, and there is no argument that this preserves normality to all bases. The proof that follows is essentially coordinate-free and would go through verbatim for any invariant interval, so the gap is removable, but as written it is a real gap. The stress-test note about this is correct. Fixing it means either removing the WLOG and working on the original interval, or proving directly that the specific affine normalization preserves absolute normality.\n\nMinor: Section 3.3 says 'we are now in position to prove Theorem 1.3' where it means Theorem 1.2. The inequality 'if N = N(ε)·c0' is garbled; it should be something like N(ε)/c0. These are easy fixes.\n\nBottom line: this is a useful survey and a readable proof of a known theorem, but the normalization gap has to be fixed before it can be taken as a rigorous self-contained proof. It deserves peer review because the expository contribution is real and the proof has enough new presentation to merit checking. I would not cite it in my own work over the original [8], but I'd send it back for revision with a request for the normalization fix, not because the theorem is in doubt.","headline":"A readable, honest survey with a self-contained proof of a known theorem; the proof is sound in substance but the opening reduction to [0,1] is not justified as written.","tokens_in":20237,"tokens_out":9943,"would_cite":false,"duration_ms":98091,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A45","28A80","11K16"],"pacs":[],"model":"deepseek-v4-flash","headline":"A self-similar measure whose Fourier transform vanishes at infinity is pointwise absolutely normal, with no rate of decay needed.","keywords":["self-similar measures","normal numbers","Rajchman property","Fourier decay","equidistribution","metric number theory","fractals","uniform distribution"],"falsifier":"Construct (or find) a non-atomic self-similar measure $\\nu$ with $F_\\xi(\\nu)\\to 0$ as $|\\xi|\\to\\infty$ but with a positive-$\\nu$ set of points that fail to be $b$-normal in some integer base $b\\ge2$. The theorem predicts no such measure exists, so a concrete example, or an explicit numerical computation of orbit statistics for a candidate measure showing non-equidistribution on a set of positive measure, would settle the claim.","tokens_in":19268,"feed_emoji":"🔢","tokens_out":9676,"duration_ms":89087,"temperature":0.7,"pith_summary":"This paper gives a self-contained proof of a recent theorem: if a non-atomic self-similar measure on the line has Fourier transform tending to zero at infinity — the Rajchman property — then almost every point in its support is absolutely normal, meaning normal in every integer base. This removes the quantitative decay rate required by the classical criterion from 1964, which is rarely verifiable for fractal measures even when decay is known qualitatively. The proof compares the statistics of base-$b$ orbits of typical points with statistics of conditional measures on a carefully chosen stopping-time partition, and shows that the Rajchman property alone forces the difference to vanish. The paper also explains how this theorem, together with a classification of Rajchman self-similar measures and a result on base-$b$ normality for non-commensurable contraction ratios, yields a structural description of the only obstructions to absolute normality: digit-like restrictions after an affine conjugation.","feed_headline":"Fourier decay alone forces full base normality on self-similar sets","feed_subtitle":"Typical points of Rajchman self-similar measures are normal to every integer base, with no decay rate required.","key_machinery":"The load-bearing mechanism is the orbit-comparison theorem stated as Theorem 2.1: for a compact space, a continuous map $T$, and a refining sequence of Borel partitions $A_k$ satisfying uniform diameter decay (2.2), the empirical orbit averages $\\frac1N\\sum_{n=1}^N \\delta_{T^n x}$ and the averages of pushed conditional measures $\\frac1N\\sum_{n=1}^N T^n \\nu_{A_n(x)}$ have the same weak-* limit for $\\nu$-almost every $x$. The proof of Theorem 1.2 applies this on the space $\\mathcal{A}^{\\mathbb{N}}\\times\\mathbb{T}$, where the partitions are cut by stopping times $\\tau_n(\\omega)$ chosen so that the derivative of the cylinder map is roughly $b^{-n}$; condition (2.2) is verified through the uniform lower bound $\\tau_n(\\omega)\\ge c n$ together with the uniform boundedness of the contraction ratios. This reduction turns base-$b$ normality into a Fourier statement: along the pushed measures, every nonzero integer frequency is multiplied by a factor bounded below by $c|E|$, so the Rajchman property forces the relevant Fourier coefficients to zero.","core_discovery":"The central claim is Theorem 1.2: every non-atomic self-similar measure $\\nu$ on $\\mathbb{R}$ with the Rajchman property — $\\widehat{\\nu}(\\xi)=\\int e^{2\\pi i \\xi x}\\,d\\nu(x)\\to 0$ as $|\\xi|\\to\\infty$ — is pointwise absolutely normal, meaning $\\nu$-almost every $x$ is normal to every integer base $b\\ge 2$. The proof given here does not require any rate of decay, which is precisely the improvement over the 1964 criterion. The argument fixes a base $b$, applies the orbit-comparison theorem to the graph of the coding map over $\\mathcal{A}^{\\mathbb{N}}\\times\\mathbb{T}$, and reduces $b$-normality to vanishing of Fourier coefficients of pushed conditional measures; the Rajchman property supplies that vanishing uniformly because the pushed frequencies are bounded below by $c|E|$. From this the paper derives a structural classification: if a self-similar measure is not pointwise absolutely normal, then after an affine conjugation its IFS has contraction ratios whose logarithms are rational multiples of $\\log b$ and translations of the form $k/b^q$ for some integer $b>1$, so the failure is caused by digit-like restrictions.","pith_inferences":["The proof relies on affine structure to shift Fourier frequencies by a factor bounded below; a natural extension would test whether vanishing Fourier decay still forces normality for non-conformal IFSs, where the same frequency-shift argument needs a nonlinear analogue.","A concrete numerical test would sample points from a Rajchman self-similar measure with overlaps and estimate base-2 digit frequencies over long blocks; the theorem predicts convergence with no exceptional digits, and the observed rate would measure how far the qualitative theorem is from an effective one.","The contrapositive suggests a sharper, unproved link: inside a self-similar measure, positive measure failure of absolute normality should force the Fourier transform to have a nonzero accumulation point at infinity, and one could investigate whether the size of the exceptional set is governed by the rate of Fourier growth."],"forward_implications":["Every Rajchman self-similar measure is pointwise absolutely normal; no decay rate is needed.","If a self-similar measure is not pointwise absolutely normal, its IFS must admit an affine conjugation under which the logarithms of the contraction ratios are rational multiples of $\\log b$ and the translations are of the form $k/b^q$ — digit restrictions are essentially the only obstruction.","If even one contraction ratio has logarithm not rationally commensurate with $\\log b$, then almost every point of the attractor is $b$-normal.","The theorem gives a route to absolutely normal numbers inside singular fractal sets with Fourier decay, including many self-similar sets with overlaps.","Effective versions of the equidistribution remain open: the martingale argument is qualitative, so rates of convergence are not supplied."],"supporting_citations":[{"why":"Supplies Theorem 2.1, the orbit-comparison theorem with martingale differences that carries the proof of Theorem 1.2.","marker":"[23]"},{"why":"First version of the comparison theorem; establishes the zoom-in/conditional-measure mechanism for uniformly scaling measures.","marker":"[24]"},{"why":"Classical criterion requiring a quantitative Fourier decay rate; Theorem 1.2 is its improvement for self-similar measures.","marker":"[17]"},{"why":"Source of Theorem 1.2, originally proved there; this paper provides the full self-contained proof.","marker":"[8]"},{"why":"Classifies which self-similar measures have the Rajchman property; used in deriving Theorem 1.1's obstruction structure.","marker":"[13]"},{"why":"Supplementary classification of Fourier decay for self-similar measures; supports the structure of obstructions in Theorem 1.1.","marker":"[29]"}],"fun_headline_variants":["Fourier decay alone makes self-similar points normal to all bases","Rajchman property gives absolute normality for self-similar measures","Self-similar + Fourier decay = normal to every base","Fourier decay forces normality to all bases on self-similar sets","Mere Fourier decay gives absolute normality for self-similar measures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the uniform shrinking condition (2.2): the diameters of the stopping-time partition cells must tend to zero uniformly after $B$ iterates of the base-$b$ map, which the proof obtains from the uniform lower bound $\\tau_n(\\omega)\\ge c n$ and from the contraction ratios of the IFS being bounded away from 1. If an IFS had contraction ratios accumulating at 1, this uniform estimate would break and the transfer argument would no longer go through.","fun_headline_variants_meta":{"raw":{"variants":["Fourier decay alone makes self-similar points normal to all bases","Rajchman property gives absolute normality for self-similar measures","Self-similar + Fourier decay = normal to every base","Fourier decay forces normality to all bases on self-similar sets","Mere Fourier decay gives absolute normality for self-similar measures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000833,"raw_usage":{"total_tokens":3610,"prompt_tokens":895,"completion_tokens":2715,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":511,"completion_tokens_details":{"reasoning_tokens":2626}},"tokens_in":511,"tokens_out":2715,"duration_ms":18498,"temperature":1.0,"reasoning_tokens":2626,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:24:40.971190+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct (or find) a non-atomic self-similar measure $\\nu$ with $F_\\xi(\\nu)\\to 0$ as $|\\xi|\\to\\infty$ but with a positive-$\\nu$ set of points that fail to be $b$-normal in some integer base $b\\ge2$. The theorem predicts no such measure exists, so a concrete example, or an explicit numerical computation of orbit statistics for a candidate measure showing non-equidistribution on a set of positive measure, would settle the claim.","supporting_citations":[{"cited_title":"A short proof of Host’s equidistributi on theorem","cited_arxiv_id":null,"evidence_quote":"Supplies Theorem 2.1, the orbit-comparison theorem with martingale differences that carries the proof of Theorem 1.2."},{"cited_title":"Equidistribution f rom fractal measures","cited_arxiv_id":null,"evidence_quote":"First version of the comparison theorem; establishes the zoom-in/conditional-measure mechanism for uniformly scaling measures."},{"cited_title":"Davenport, P","cited_arxiv_id":null,"evidence_quote":"Classical criterion requiring a quantitative Fourier decay rate; Theorem 1.2 is its improvement for self-similar measures."},{"cited_title":"Po intwise normality and Fourier decay for self-conformal measures","cited_arxiv_id":null,"evidence_quote":"Source of Theorem 1.2, originally proved there; this paper provides the full self-contained proof."},{"cited_title":"Self-similar measures and the Rajchman property","cited_arxiv_id":null,"evidence_quote":"Classifies which self-similar measures have the Rajchman property; used in deriving Theorem 1.1's obstruction structure."},{"cited_title":"Trigonometric series and self-similar sets","cited_arxiv_id":null,"evidence_quote":"Supplementary classification of Fourier decay for self-similar measures; supports the structure of obstructions in Theorem 1.1."}],"review_version":1}