{"id":"d07f2b00-4c3b-47e7-8ba6-abb51851b415","arxiv_id":"2506.22192","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New mean value estimates for exponential sums over y-smooth numbers with y=(log x)^{K+o(1)} give a power saving over the trivial bound in an intermediate range, and yield a nontrivial bound on additive energy for K>12.","lead":"This paper proves new upper bounds for the average size of exponential sums over numbers with no prime factor above (log x)^K, a sparse but structured set. The bounds improve on the trivial estimate in a range of K and rho that earlier theorems did not cover, including a power saving for the additive energy of such smooth numbers.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central saving rests on unverified simplified forms of Harper/Baker pointwise bounds; if Lemma 2.3's exponents or support are misquoted, Corollary 1.4's zeta changes.","rationale":"The reader's weakest assumption is exactly the one I find load-bearing: the simplified pointwise bounds of Lemmas 2.2 and 2.3 are quoted from Harper and Baker and not re-derived. Because Theorem 1.3's optimal Q, the exponent eta, and hence Corollary 1.4's zeta are all computed from those pointwise inequalities, a transcription error in either lemma would propagate directly into the headline corollary. I independently verified the internal algebra: Theorem 1.1 follows from (3.11) and (3.12); Theorem 1.3's balancing equation (4.7) gives xi = 1 - 2eta with Q = x^xi >= x^(1/2) equivalent to rho < 2K + 4; and for rho = 4 the bound E <= Psi^(1+o(1)) x^((2+9kappa)/(1+6kappa)) equals Psi^(3-zeta+o(1)) with zeta = kappa(1-12kappa)/(1+5kappa-6kappa^2). I also noted a display typo in the Corollary 1.4 derivation: the intermediate exponent is printed with denominator 1+5kappa-6kappa^2 instead of 1+6kappa, but the final zeta is consistent with the corrected denominator, so this is not a mathematical defect. The remaining uncertainty is purely whether the quoted simplified external bounds are faithful; since these are published results, a targeted verification settles the concern, and acceptance should be conditional on that check.","tokens_in":10262,"tokens_out":11936,"duration_ms":115463,"concrete_test":"Retrieve Harper [18, Theorem 1] and Baker [2, Theorem 2] and translate parameters, using Baker's kappa as 1 - kappa in the paper's notation. Verify that Lemma 2.2's bound holds for every qL <= 2 x^(1/2-epsilon) with the stated exponent gamma = (1-3kappa)/2, and that Lemma 2.3's bound holds with no additional restriction on qL and with exactly the second term (qL x^(kappa-1))^(1/2). If both match, Corollary 1.4 follows; if the exponents or support differ, recompute (4.6)-(4.7) with the correct pointwise bound and check whether E(x,y) still admits a power-of-x saving for K > 12.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 1.3 reduces the mean value to the terms G7, H7, G5, H5 in (4.5) using only the simplified pointwise bounds Lemma 2.2 (Harper) and Lemma 2.3 (Baker). In particular, (4.3)-(4.4) assume that uniformly for all q <= x/Q and L = 1 + x|theta - a/q| one has |S(theta;x,y)| <= Psi^(1+o(1)) ((qL)^(-gamma) + (qL x^(kappa-1))^(1/2)) with gamma = (1-3kappa)/2. The saving exponent zeta in Corollary 1.4 comes from balancing (4.6) against (4.2); if Baker's theorem actually has an additional restriction on qL, or if its second term carries a different power of x, then the bounds on H7/H5 and the balancing exponent xi in (4.7) change, and the stated power saving may shrink or disappear. The paper labels these lemmas as simplified and does not re-prove them, so the central claim is only as secure as the faithful transcription of [18, Theorem 1] and [2, Theorem 2]. I checked the internal algebra of Sections 3-4 under those lemmas and found no further flaw; the issue is the external premise, not the derivation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves two mean value theorems for exponential sums over y-smooth numbers up to x in the range y=(log x)^{K+o(1)}. Theorem 1.1 gives a bound for I_rho(x,y) when K>3 and beta rho>1/2, with a nontrivial range rho>4(K-1)/(K-3), K>4. Theorem 1.3 targets the additive energy case and, after balancing major-arc and minor-arc contributions, yields Corollary 1.4: for K>12, E(x,y) <= Psi(x,y)^{3-zeta+o(1)} with zeta = kappa(1-12kappa)/(1+5kappa-6kappa^2) and kappa=1/K. The proofs use Dirichlet approximation, the Parseval identity, and pointwise bounds quoted from Fouvry-Tenenbaum (Lemma 2.1), Harper (Lemma 2.2), and Baker (Lemma 2.3).","tokens_in":10454,"tokens_out":47007,"duration_ms":453433,"significance":"If correct, the paper supplies the first power savings for the additive energy of (log x)^K-smooth numbers in the intermediate range K>12, interpolating between the small-y results of Bourgain-Garaev-Konyagin-Shparlinski and the large-y mean value theorem of Harper. The internal algebra of Sections 3 and 4 is consistent, there are no fitted parameters, and the dependence on the quoted external pointwise bounds is explicit. The main caveat is that the final saving exponent zeta is directly computed from the simplified forms of Harper's and Baker's theorems in Lemmas 2.2 and 2.3; I found no demonstrated misquotation, but the manuscript would be easier to certify if those forms were located precisely in the cited sources. The paper is clearly written and the claimed result is concrete and falsifiable.","major_comments":[],"minor_comments":[{"comment":"The first sentence says 'We proceed as in the proof of Theorem 1.3', but the surrounding argument is a variant of the proof of Theorem 1.1; this should be corrected.","section":"Section 4.1"},{"comment":"The phrase 'we now chose Q' should read 'we now choose Q'.","section":"Section 4.3"},{"comment":"The sentence 'which follows from the Parseval identity I2(x,y)=Psi(x,y)' is duplicated verbatim in the displayed text; the duplicate should be removed.","section":"Section 1.1"},{"comment":"The step replacing the sum over Farey arcs by I2(x,y), namely 'sum_{q,a} int_{M_{a,q,Q}} |S|^2 dtheta ! I2(x,y)', should be justified, because the arcs M_{a,q,Q} overlap. With the standard bounded-overlap (up to O(log Q)) property this is harmless and is absorbed by the x^{o(1)} factor, but the justification should be stated explicitly.","section":"Sections 3.3 and 4.2"},{"comment":"Since the exponent gamma=(1-3kappa)/2 and the second term (qL x^{kappa-1})^{1/2} in Lemma 2.3 directly determine the saving exponent zeta in Corollary 1.4, the authors should add a short note confirming that these simplified forms follow from [18, Theorem 1] and [2, Theorem 2] with no additional hidden restrictions on qL or on the range of theta. This would remove a genuine verification burden on the reader.","section":"Section 2.2, Lemmas 2.2 and 2.3"}],"recommendation":"minor_revision","confidential_remarks":"The stress-test concern about the simplified external bounds in Lemmas 2.2 and 2.3 is a legitimate caution but did not materialize as an internal inconsistency: the algebra in Sections 3 and 4 is consistent under the stated lemmas. I recommend minor revision mainly to add the requested verification notes and to fix the small presentation issues. If the authors confirm the faithful transcription of [18, Theorem 1] and [2, Theorem 2], I would support acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one if you care about mean values of exponential sums over y-smooth numbers in the intermediate range y = (log x)^K. The paper does exactly what it says: it takes the simplified pointwise bounds of Harper, Baker, and Fouvry–Tenenbaum, runs them through a Dirichlet partition, and produces mean value theorems that are nontrivial in ranges not covered by the earlier small-y or large-y results. The most concrete payoff is Corollary 1.4, a power-of-x saving for the additive energy of y-smooth numbers when K > 12. That corollary is new and genuinely follows from the main theorem.\n\nWhat I like: the paper is transparent about its structure. It states the nontriviality conditions explicitly, the balancing of terms in Sections 3–4 is internally consistent, and the authors do not oversell. They flag that the pointwise bounds are simplified from the literature and they do not claim to have re-proved them. The citation pattern is appropriate, with no fitted parameters and no circularity. The internal algebra checks out; I followed the derivation with beta = (1 – 3κ)/4 and gamma = 2β, and the thresholds in (1.7) and (1.8) line up.\n\nThe soft spot is exactly the one the stress-test note identifies: the entire saving exponent zeta in Corollary 1.4 depends on the precise form of Lemma 2.2 (Harper) and Lemma 2.3 (Baker), especially the exponent of (qL) and the condition (2.2). If Baker's theorem actually carries an additional restriction on qL, or if his second term has a different x-power, then the H7/H5 bounds and the balancing exponent xi in (4.7) shift, and the stated power saving could shrink or vanish. This is not a flaw in the paper's derivation, because the lemmas are quoted from published work and the authors are upfront about simplification. But it is load-bearing, and a referee should check those transcriptions against the originals before trusting Corollary 1.4 quantitatively. The paper would not collapse if a minor exponent changed—the general method would survive—but the exact statement might need adjustment.\n\nMinor issues: a couple of typos and a duplicated sentence in Section 1.1, but nothing that obscures the mathematics.\n\nWho is this for? Number theorists working on smooth numbers, mean values, and additive energy. It is a modest but real advance, not a breakthrough. I would send it to a competent referee rather than desk-reject it, and I would expect the referee's main job to be verification of the quoted external bounds.\n\nRecommendation: accept after a check of Lemmas 2.2 and 2.3 against Harper's and Baker's theorems. The paper deserves referee time.","headline":"A solid, honest extension of existing pointwise bounds to new mean value ranges; the main risk is the unverified transcription of Harper's and Baker's bounds, but the internal derivation is clean and the paper merits serious refereeing.","tokens_in":11135,"tokens_out":1714,"would_cite":false,"duration_ms":21168,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11L07","11N25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For K>12, the additive energy of smooth numbers saves a power of x over the trivial bound.","keywords":["smooth numbers","exponential sums","mean value theorems","additive energy","power saving","sparse sets","Dirichlet approximation"],"falsifier":"A direct check for $K=13$ and increasing $x$ on the major arcs $qL\\le 2x^{1/2-\\varepsilon}$ would test whether $|S(\\theta;x,y)| \\le \\Psi(x,y)^{1+o(1)}(qL)^{-\\gamma}(\\log x)^{5/2+o(1)}$ holds; finding a single family of $\\theta$ where the right-hand side is exceeded by a power of $\\log x$ would invalidate the main saving and force a recomputation of $\\zeta$ in Corollary 1.4.","tokens_in":9954,"feed_emoji":"🔢","tokens_out":13659,"duration_ms":126491,"temperature":0.7,"pith_summary":"The paper proves new mean value bounds for exponential sums over $y$-smooth numbers when $y$ grows like a power of $\\log x$, the range where the set is sparse ($\\Psi(x,y)=x^{1-\\kappa+o(1)}$) but where no previous bound improved on the trivial estimate $\\Psi^{\\rho-1}$. Its main theorem gives, for admissible $\\rho$ and $K$, $I_\\rho(x,y) \\le \\Psi(x,y)^{1+o(1)}(x^{3(\\rho-2)/4}+x^{(\\rho-2)(1-\\eta)})$, with an explicit $\\eta$. The corollary for $\\rho=4$ is a power saving for the additive energy: for $K>12$, $E(x,y) \\le \\Psi(x,y)^{3-\\zeta+o(1)}$ with $\\zeta=\\kappa(1-12\\kappa)/(1+5\\kappa-6\\kappa^2)$. This closes part of the gap between the very small $y$ where an optimal bound is known and the larger $y$ where a different pointwise method already works.","feed_headline":"Smooth-number sums get power saving in a new range","feed_subtitle":"For y=(log x)^K and K>12, the additive energy falls below the trivial Ψ³ by a power of x.","key_machinery":"The proof splits $[0,1]$ by Dirichlet approximation into arcs $M_{a,q,Q}$ centered at $a/q$ with $q\\le Q$. On arcs with $q\\le x^{1/2-\\varepsilon}$ it uses a pointwise bound (Lemma 2.2) giving $|S(\\theta;x,y)| \\le \\Psi^{1+o(1)}(qL)^{-\\gamma}(\\log x)^{5/2+o(1)}$ whenever $qL\\le 2x^{1/2-\\varepsilon}$; on all arcs it uses an unconditional variant (Lemma 2.3) with an extra $(qL x^{\\kappa-1})^{1/2}$ term, where $L=1+x|\\theta-a/q|$ and $\\gamma=(1-3\\kappa)/2$. The large-denominator contribution is controlled by a third pointwise bound (Lemma 2.1). Balancing the two main terms at $Q=x^\\xi$ with $\\xi=1-2\\eta$ converts the pointwise decay into the mean-value exponent $(\\rho-2)(1-\\eta)$.","core_discovery":"Let $S(\\theta;x,y)=\\sum_{n\\in S(x,y)}e^{2\\pi i \\theta n}$ be the exponential sum over the $y$-smooth numbers up to $x$. The paper establishes that for $y=(\\log x)^{K+o(1)}$ with $K>3$, whenever $1/2<\\beta\\rho<1$ and $2<\\rho<2K+4$, the $\\rho$-th moment satisfies the bound stated in Theorem 1.3, and that the choice $\\rho=4$, $K>12$ yields $E(x,y)=I_4(x,y) \\le \\Psi(x,y)^{3-\\zeta+o(1)}$. Here $\\kappa=1/K$, $\\beta=(1-3\\kappa)/4$, $\\eta=\\kappa(\\rho-1)/(2+3\\kappa\\rho)$, and $\\zeta=\\kappa(1-12\\kappa)/(1+5\\kappa-6\\kappa^2)$. Because $\\zeta>0$ exactly when $K>12$, the fourth moment is smaller than the trivial $\\Psi^3$ by a factor $\\Psi^{-\\zeta}$, which is a power of $x$ since $\\Psi=x^{1-\\kappa+o(1)}$. The paper presents this as the first power saving in this intermediate range.","pith_inferences":["A sharper version of the quoted pointwise bounds could push the threshold in Corollary 1.4 from K>12 toward K>4, the natural boundary where the trivial bound currently takes over; testing this would require revisiting the two imported lemmas rather than the arc-splitting argument itself.","The exponent ζ has the form κ(1−12κ)/(1+5κ−6κ^2), and a matching lower bound is not addressed: one could try to construct many distinct solutions to n1+n2=n3+n4 with all ni y-smooth to see whether the energy is forced to stay close to Ψ^3 for K just above 12.","Applying the same balancing at Q=x^{1/2} exactly, rather than with the ε-slack used here, might replace the o(1) exponent by a computable logarithmic factor and make the bound directly comparable with numerical data for moderate x.","The method could be extended to weighted smooth numbers or to products of smooth numbers; the arc-splitting would survive, but the pointwise inputs would need to be re-verified."],"forward_implications":["For K>12 the additive energy of the (log x)^K-smooth numbers up to x is bounded by Ψ(x,y)^{3−ζ+o(1)}, a power saving over the trivial Ψ^3.","The same theorem gives nontrivial mean value bounds for every even ρ in the stated range, so the method applies beyond the fourth moment.","The bound on E(x,y) feeds into the known criterion connecting additive energy to the metric Poissonian pair-correlation property, extending the range of K for which smooth numbers have that property.","The argument also yields estimates for the number of solutions to linear equations and for matrix counts with entries from the smooth set, as outlined in the applications section.","The new bounds interpolate between the regime of very small y where an optimal bound is known and the regime of larger y where a different method already gave power savings."],"supporting_citations":[{"why":"Supplies the pointwise bound (Lemma 2.2) on smooth-number exponential sums used on small-denominator arcs; its quoted form carries the main saving.","marker":"[18]"},{"why":"Supplies the unconditional pointwise bound (Lemma 2.3) with the additional square-root term used for Theorem 1.3.","marker":"[2]"},{"why":"Supplies the elementary pointwise bound (Lemma 2.1) that controls the large-denominator contribution.","marker":"[15]"},{"why":"Gives the asymptotic Ψ(x, log^K x)=x^{1−κ+o(1)} used to express the bounds as powers of x.","marker":"[17]"},{"why":"Gives the saddle-point behaviour α(x,y)=1−κ+o(1) used to state and simplify the pointwise bounds.","marker":"[19]"}],"fun_headline_variants":["Smooth-number sums get power saving for K>12","Exponential sums over smooth numbers beat trivial bound","New range of power saving for smooth-number sums","Fourth moment of smooth sums beats trivial by power","Mean value theorems for smooth numbers reach further"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof inherits the two pointwise bounds on $|S(\\theta;x,y)|$ with exactly the stated exponents, and one of them only under the restriction $qL\\le 2x^{1/2-\\varepsilon}$; if those quoted bounds are not valid in this form, the saving in the mean values has to be reworked.","fun_headline_variants_meta":{"raw":{"variants":["Smooth-number sums get power saving for K>12","Exponential sums over smooth numbers beat trivial bound","New range of power saving for smooth-number sums","Fourth moment of smooth sums beats trivial by power","Mean value theorems for smooth numbers reach further"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000386,"raw_usage":{"total_tokens":1976,"prompt_tokens":817,"completion_tokens":1159,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":433,"completion_tokens_details":{"reasoning_tokens":1087}},"tokens_in":433,"tokens_out":1159,"duration_ms":12728,"temperature":1.0,"reasoning_tokens":1087,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:12:52.954820+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check for $K=13$ and increasing $x$ on the major arcs $qL\\le 2x^{1/2-\\varepsilon}$ would test whether $|S(\\theta;x,y)| \\le \\Psi(x,y)^{1+o(1)}(qL)^{-\\gamma}(\\log x)^{5/2+o(1)}$ holds; finding a single family of $\\theta$ where the right-hand side is exceeded by a power of $\\log x$ would invalidate the main saving and force a recomputation of $\\zeta$ in Corollary 1.4.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the pointwise bound (Lemma 2.2) on smooth-number exponential sums used on small-denominator arcs; its quoted form carries the main saving."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the unconditional pointwise bound (Lemma 2.3) with the additional square-root term used for Theorem 1.3."},{"cited_title":"Fouvry and G","cited_arxiv_id":null,"evidence_quote":"Supplies the elementary pointwise bound (Lemma 2.1) that controls the large-denominator contribution."},{"cited_title":"Granville, ‘Smooth numbers: Computational number theory and beyond’, Proc","cited_arxiv_id":null,"evidence_quote":"Gives the asymptotic Ψ(x, log^K x)=x^{1−κ+o(1)} used to express the bounds as powers of x."},{"cited_title":"Hildebrand and G","cited_arxiv_id":null,"evidence_quote":"Gives the saddle-point behaviour α(x,y)=1−κ+o(1) used to state and simplify the pointwise bounds."}],"review_version":1}