{"id":"0d9c9a58-b785-4bba-8ef3-f1c596b6094d","arxiv_id":"2412.16621","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Self-similar measures on self-similar sets are proved to have explicit power-of-logarithm Fourier decay, and the result is applied to Lüroth sets with restricted digits.","lead":"This paper derives explicit formulas for how fast Fourier transforms of self-similar measures on self-similar sets decay, sharpening an earlier logarithmic decay result whose exponent was only implicit. The formulas are applied to fractal sets defined by Lüroth expansions with digit restrictions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3's proof verifies weak diophantineness only on a lattice of frequencies, not the full real line; this gap sits between the non-Liouville hypothesis and the renewal estimate used in Theorem 1.","rationale":"I read the paper in good faith. The main lines of Theorem 1 are coherent: Propositions 1, 2, 5 and the t-balancing in (21) are consistent, and the final exponent follows. The strongest concrete flaw I find is in Proposition 3: the proof establishes the weak diophantine property on a discrete sequence, not on the real line, despite the definition requiring a liminf. Because Proposition 4 is imported from [8] and invoked at all frequencies, this is the load-bearing bridge from the arithmetic hypothesis to the renewal estimate. The reader's flagged concern about Proposition 9's Matveev constants is also legitimate but only affects the Lüroth application (Theorem 4), not Theorem 1. I therefore partially agree with the reader, and I keep the CONDITIONAL verdict: the central claim is likely correct, but the proof as written has an unverified step that should be completed before acceptance.","tokens_in":24520,"tokens_out":41964,"duration_ms":345563,"concrete_test":"Analytically settle the missing perturbation step: for arbitrary τ, write n=round(τ log r_b/2π) and ε=τ-2πn/log r_b, and prove that (w_a/2)|1-e^{-2πiθn}e^{-iaε}|^2+(w_b/2)|1-e^{-ibε}|^2 ≥ c(||θn||^2+||ε||^2) for all large n, with c>0 depending only on λ, θ and the non-Liouville constant. If this inequality holds, Proposition 3 is correct and only the exposition needs revision; if it fails, the weak diophantineness claim and hence Theorem 1's off-diagonal estimate collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition of weakly diophantine requires liminf over all τ. The proof of Proposition 3 picks τ1∈Z\\{0}, sets τ=2πτ1/log r_b, and derives |τ|^{2s-2}|1-L_λ(iτ)|≫1. This confirms the bound on the arithmetic progression (2π/log r_b)Z, but not for τ off this lattice. Proposition 4's renewal estimate is then applied in Proposition 5 to all frequencies arising from the Fourier inversion, so the full liminf is needed. As written, Proposition 3's conclusion does not follow. The gap is concrete: for τ=2πn/log r_b+ε, the real part of 1-L_λ(iτ) equals (w_a/2)|1-e^{-2πiθn}e^{-iaε}|^2+(w_b/2)|1-e^{-ibε}|^2; proving the stated claim requires a lower bound of this by c(||θn||^2+ε^2) (up to constants), which the manuscript does not supply. A standard perturbation argument can likely fill this, but the text is incomplete.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an explicit logarithmic decay rate for Fourier transforms of self-similar probability measures on self-similar sets in [0,1], assuming that the ratio of two contraction logarithms is non-Liouville of degree s. Theorem 1 states that for any upper regularity exponent alpha of the measure, |mu-hat(xi)| = O(log^{-alpha/(2(1+2alpha)(8s-7))}|xi|) as |xi| tends to infinity. The proof combines a diagonal estimate using upper regularity with an off-diagonal estimate obtained from a quantitative renewal theorem, after verifying that an auxiliary measure is weakly diophantine. Theorem 2 gives a new proof of Moran's dimension formula, Theorem 3 constructs a self-similar measure that attains the fastest decay rate allowed by Theorem 1 under an open-set-type condition, and Theorem 4 applies the machinery to sets of numbers with digits restricted in their Luroth expansions, using Matveev's explicit Baker theorem to verify the non-Liouville condition.","tokens_in":24725,"tokens_out":8991,"duration_ms":79401,"significance":"If the proof is completed, the paper makes a genuine contribution: it converts the implicit logarithmic decay rate of Li-Sahlsten into an explicit, parameter-free exponent depending only on the upper regularity exponent and the non-Liouville degree. The construction in Theorem 3 of a measure achieving the maximal rate is a useful and nontrivial addition, and the application to restricted Luroth digits gives a fully explicit, though numerically small, decay exponent. The new proof of Moran's formula via upper regularity exponents is a legitimate alternative route. The main claims are quantitative and falsifiable, and the paper does not use fitted parameters or circular reasoning; its dependence on prior results by Li-Sahlsten, Li, and Matveev is clearly stated.","major_comments":[{"comment":"Proposition 4 is stated for C^1 functions f : R -> [0,+infty), but in Proposition 5 it is applied to the complex-valued function f_theta(s) = exp(-2 pi i theta e^{-s}) (equation (23)). The renewal estimate is linear, so the application can be repaired by splitting into real and imaginary parts or by stating a complex-valued version, but as written the theorem statement does not cover the case actually used.","section":"Section 2, Proposition 4 and its use in Proposition 5"}],"minor_comments":[{"comment":"In the last paragraph of the proof, the formula for tau changes from 2 pi tau_1 / log r_b to 2 pi tau_1 / log r_a; the two denominators should be reconciled.","section":"Section 2, proof of Proposition 3"},{"comment":"The sentence defining t as the unique solution to equation (21) would benefit from an explicit remark that existence and uniqueness hold for all sufficiently large |xi|; this is implicit in the proof but not stated.","section":"Section 2, Proposition 5"},{"comment":"The passage from beta_I,0 to the simpler expression beta_I uses the assertion 'beta_I,0 >= beta_I'; a one-line verification of this inequality would help the reader confirm the numerical claim.","section":"Section 5, Theorem 4 and Proposition 10"},{"comment":"The statement allows the degenerate case alpha = 0, which gives only the trivial O(1) bound; a brief remark that this is consistent with the Frostman lemma and the singleton case would be useful.","section":"Section 1, Theorem 1"}],"recommendation":"major_revision","confidential_remarks":"The main gap is the incomplete verification of weak diophantineness in Proposition 3. I believe it is repairable with a perturbation argument, and the rest of the strategy is coherent. The paper fits the scope of math.CA and the application to Luroth sets is a concrete dividend. The author should also double-check the constants in the Matveev-based lower bound against the source, since the reported numerical bound '>10^{-11}' depends on them."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid, honest paper that does exactly what the title says--makes the decay exponent in Li-Sahlsten's theorem explicit. The main result, Theorem 1, gives a log^{-alpha/(2(1+2alpha)(8s-7))} bound, and the proof is mostly careful parameter-tuning of the Li-Sahlsten renewal machinery rather than a new mechanism. That is fine; the paper says so.\n\nThe genuinely new pieces: the explicit formula, the construction in Theorem 3 of a measure achieving the exponent with alpha = dim E, and the Luroth application in Theorem 4. I checked the algebra in Propositions 2 and 5 and it is consistent; the choice of t in (21) gives the claimed exponent. The paper is transparent about Moran's theorem, and the new proof of Moran's formula using the measure nu_alpha is clean. No fitted parameters, no circularity. Credit where it is due: this is an honest incremental advance.\n\nSoft spots, in decreasing order of seriousness. First, Proposition 3 only verifies weak diophantineness on the arithmetic progression (2pi/log r_b)Z. The definition needs the liminf over all real tau. The missing perturbation argument--bounding the off-lattice frequencies by combining the non-Liouville distance with the epsilon^2 contribution from the b-term--should work, but it is not in the text. As written the proof has a genuine gap, so the theorem is conditional on that being filled. Second, Proposition 9's use of Matveev's theorem does not read cleanly: the displayed computation appears to gain an extra factor in the exponent, and the numerical constants are not re-derivable from the cited theorem as written. This deserves a careful check. Third, and minor: the exponents are extremely small (best case 1/54), and the paper honestly notes it does not reach the PVZZ threshold. That limits the payoff but is not a flaw.\n\nThe paper is for people working on Fourier decay of self-similar measures and quantitative metric number theory. It deserves a serious referee. I would send it out, with requests to fix Proposition 3 and double-check Proposition 9 against Matveev; the main argument is likely sound.","headline":"A careful, honest paper that makes Li-Sahlsten's logarithmic decay exponent explicit; the main proof structure is sound, but Proposition 3 and Proposition 9 need fixing before publication.","tokens_in":682,"tokens_out":841,"would_cite":false,"duration_ms":49913,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["28A80","42A38","60K05","11K55"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every self-similar measure on a suitably arithmetic self-similar set gets an explicit logarithmic Fourier decay rate.","keywords":["Fourier transforms of measures","self-similar measures","self-similar sets","Rajchman property","quantitative renewal theory","non-Liouville numbers","L\\\"uroth expansion","upper regularity exponent"],"falsifier":"Take the L\\\"uroth set with digits $\\{2,3\\}$ and its optimal self-similar measure $\\mu_3$; compute $|\\hat{\\mu}_3(\\xi)|$ numerically at frequencies $\\xi=e^n$ for large $n$ and compare the envelope with $C\\log^{-\\beta_3}\\xi$ for the paper's $\\beta_3>10^{-11}$. If the measured decay along this sequence is slower than every positive power of $\\log|\\xi|$, or if the envelope exceeds the predicted bound by an unbounded factor, the claimed universal rate would be refuted.","tokens_in":24277,"feed_emoji":"📉","tokens_out":10360,"duration_ms":86429,"temperature":0.7,"pith_summary":"For any self-similar measure on a self-similar set whose contraction ratios satisfy a non-Liouville condition, the Fourier transform decays like a power of $\\log|\\xi|$, with the exponent explicitly given in terms of the measure's upper regularity exponent and the degree of the non-Liouville ratio. The paper thereby upgrades an earlier result whose decay exponent was implicit and unusable for quantitative applications. This matters because metric number theory sometimes requires checking that a decay exponent exceeds a concrete threshold, which was impossible before. The paper also constructs a self-similar measure attaining the fastest exponent available under the open set condition, and verifies the arithmetic hypothesis for sets of numbers with digits restricted to a finite set in their L\\\"uroth representation.","feed_headline":"Self-similar measures get explicit Fourier decay rates","feed_subtitle":"A previously implicit decay exponent is now computed from the measure's regularity and the ratio's arithmetic type.","key_machinery":"The load-bearing mechanism is a quantitative renewal theorem for the stopping time of a random walk whose increments are $-\\log r_i$ with probabilities $p_i$. The auxiliary step distribution must be weakly diophantine, meaning $|t|^{2s-2}|1-\\mathcal{L}\\lambda(it)|$ stays bounded away from zero; Proposition 3 derives this from the non-Liouville condition on $\\log r_i/\\log r_j$. Proposition 4, a renewal estimate with explicit error $O(t^{-1/(8s-7)})$, then controls the off-diagonal part of the two-dimensional integral in Proposition 1, while the diagonal part is bounded using the upper regularity exponent $\\alpha$. Balancing the two terms at a scale $t$ coupled to $|\\xi|$ by $t^{(1+\\alpha)/((1+2\\alpha)(8s-7))} e^t = |\\xi|$ produces the final logarithmic exponent.","core_discovery":"Theorem 1 is the paper's central assertion: if $\\log r_i/\\log r_j$ is non-Liouville of degree $s\\ge 2$ for some two similitudes of a self-similar set $E$, then every self-similar probability measure $\\mu$ on $E$ satisfies $|\\hat{\\mu}(\\xi)|=O(\\log^{-\\alpha/(2(1+2\\alpha)(8s-7))}|\\xi|)$ as $\\xi\\to\\pm\\infty$, for every upper regularity exponent $\\alpha$ of $\\mu$. The proof splits the Fourier integral into a diagonal region controlled by $\\alpha$ and an off-diagonal region controlled by a quantitative renewal estimate for the random walk with step distribution given by the logarithmic contraction ratios. Under the open set condition, the paper's Theorem 3 shows that the measure with weights $r_i^{\\dim E}$ has $\\alpha=\\dim E$, so the fastest rate available from this method is $O(\\log^{-\\dim E/(2(1+2\\dim E)(8s-7))}|\\xi|)$. For restricted L\\\"uroth digit sets the needed non-Liouville hypothesis is checked by an explicit lower bound on linear forms in logarithms, and the resulting positive decay exponent is spelled out.","pith_inferences":["If the non-Liouville hypothesis is dropped, the renewal estimate loses its handle; it is an editorial guess that some self-similar measures with a Liouville log-ratio will exhibit no positive logarithmic decay, and exhibiting such an example would mark the true boundary of the theorem.","The explicit constants are so small in the L\\\"uroth instance that the practical route to thresholds like $\\beta>2$ is likely to run through sharper Baker-type bounds and sharper renewal error terms, rather than through the present architecture.","The same template should transfer to other self-similar digit systems---continued-fraction Cantor sets, $\\beta$-expansions, and other L\\\"uroth-type bases---wherever the contraction ratios are algebraic and explicit lower bounds for linear forms in logarithms are available.","The role of the open set condition seems to enter only through the regularity exponent $\\alpha$; the Fourier estimate itself is independent of separation, so a different way of establishing a large $\\alpha$ could produce fast decay without the separation condition."],"forward_implications":["Every self-similar measure on such a set has a computable, positive logarithmic decay rate, so the earlier implicit parameter becomes an explicit function of the measure and the set.","Under the open set condition, the optimal measure is the one with weights $p_i = r_i^{\\dim E}$, and its decay exponent is $\\dim E/(2(1+2\\dim E)(8s-7))$, with $\\dim E$ the unique solution of $\\sum_i r_i^q=1$.","For finite restricted L\\\"uroth digit sets, there is an explicit self-similar measure with decay like $\\log^{-\\beta}|\\xi|$, where $\\beta$ is written in terms of the two smallest digits; for digits $\\{2,3\\}$, $\\beta>10^{-11}$.","Because the decay is logarithmic with positive exponent, the measure satisfies the summability condition in the classical uniform-distribution criterion, so for $\\mu$-almost every $x$ any lacunary sequence $(n_k x \\bmod 1)$ is uniformly distributed.","The explicit exponent still falls short of the threshold $\\beta>2$ needed in the stated inhomogeneous Diophantine approximation application; the paper's own best case gives $\\beta=1/54$."],"supporting_citations":[{"why":"Supplies the quantitative renewal framework and the prior implicit logarithmic-decay theorem that this paper makes explicit.","marker":"[8]"},{"why":"Provides the explicit Baker-type lower bound for linear forms in logarithms used to verify the non-Liouville condition in the L\\\"uroth application.","marker":"[11]"},{"why":"Is the source of the oscillation-integral estimate used in Proposition 5's main term.","marker":"[7]"},{"why":"Supplies the classical Moran equation that Theorem 2 reproves to identify $\\dim E$.","marker":"[12]"},{"why":"Guarantees that non-singleton self-similar measures have a positive upper regularity exponent.","marker":"[2]"},{"why":"Provides existence of the self-similar set and the self-similar measure from given similitudes and weights.","marker":"[5]"}],"fun_headline_variants":["Self-similar measures get explicit Fourier decay rates","Explicit Fourier decay bounds on self-similar sets","Sharper explicit decay for self-similar measures","New explicit decay rate for fractal Fourier transforms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the logarithm of one contraction ratio divided by the logarithm of another is a non-Liouville number of some finite degree; if that fails, the auxiliary measure is not provably weakly diophantine and the quantitative renewal estimate collapses.","fun_headline_variants_meta":{"raw":{"variants":["Self-similar measures get explicit Fourier decay rates","Explicit Fourier decay bounds on self-similar sets","Sharper explicit decay for self-similar measures","New explicit decay rate for fractal Fourier transforms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000698,"raw_usage":{"total_tokens":3115,"prompt_tokens":871,"completion_tokens":2244,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":2185}},"tokens_in":487,"tokens_out":2244,"duration_ms":15132,"temperature":1.0,"reasoning_tokens":2185,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:25:29.568521+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the L\\\"uroth set with digits $\\{2,3\\}$ and its optimal self-similar measure $\\mu_3$; compute $|\\hat{\\mu}_3(\\xi)|$ numerically at frequencies $\\xi=e^n$ for large $n$ and compare the envelope with $C\\log^{-\\beta_3}\\xi$ for the paper's $\\beta_3>10^{-11}$. If the measured decay along this sequence is slower than every positive power of $\\log|\\xi|$, or if the envelope exceeds the predicted bound by an unbounded factor, the claimed universal rate would be refuted.","supporting_citations":[{"cited_title":"F/e.sc/n.sc/g.sc /a.sc/n.sc/d.sc K./hyphen.scS","cited_arxiv_id":null,"evidence_quote":"Supplies the classical Moran equation that Theorem 2 reproves to identify $\\dim E$."},{"cited_title":"Let  be a non-empty finite set","cited_arxiv_id":null,"evidence_quote":"Supplies the quantitative renewal framework and the prior implicit logarithmic-decay theorem that this paper makes explicit."},{"cited_title":"B/e.sc/r.sc/e.sc/s.sc/n.sc/e.sc/v.sc/i.sc/c.sc/h.sc, F","cited_arxiv_id":null,"evidence_quote":"Provides the explicit Baker-type lower bound for linear forms in logarithms used to verify the non-Liouville condition in the L\\\"uroth application."},{"cited_title":"A few definitions and propositions are presented before starting the main proof of Theorem 1","cited_arxiv_id":null,"evidence_quote":"Is the source of the oscillation-integral estimate used in Proposition 5's main term."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Guarantees that non-singleton self-similar measures have a positive upper regularity exponent."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides existence of the self-similar set and the self-similar measure from given similitudes and weights."}],"review_version":1}