{"id":"28697520-13f1-4716-915d-937c34ac11e9","arxiv_id":"2608.08893","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Fixed-radius lattice sphere averages on Z^d satisfy the sharp l^p improving estimate down to the endpoint p=(d+2)/d for every d at least 4, with the optimal decay exponent.","lead":"This paper proves the exact range of exponents in which averaging a function over lattice points at a fixed distance must shrink its size by the optimal amount. It completes a problem in discrete harmonic analysis that previous work had narrowed down but not fully solved.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Uniform arbitrary-center embedded-sphere lattice bound (Lemma 2.1) is load-bearing: if it carries any center/determinant factor, the low-density endpoint (6.6)–(1.9) fails.","rationale":"The reader's weakest assumption exactly matches the step I consider most load-bearing: the uniform arbitrary-center embedded-sphere lattice count. The proof of Theorem 1.1 routes the whole low-density regime through Lemma 6.2, and Lemma 6.2 inherits all of its quantitative content from Lemma 2.1. A failure or weakening of that lemma is not a minor technicality; it changes the power of R in (6.6) and destroys the desired endpoint. The reader's other concern, the omitted proof of Lemma 3.1, is real but lower-risk: it is a routine divisor sum, whereas Lemma 2.1 is a deep uniformity statement that is not reproduced. I therefore keep the existing CONDITIONAL verdict rather than moving to accept, reject, or unverified: the central claim is plausible and the proof is coherent, but it should not be accepted as fully verified until Lemma 2.1's exact hypotheses are confirmed or supplied.","tokens_in":17140,"tokens_out":26845,"duration_ms":299583,"concrete_test":"Verify the exact statement of Huang–Zhang [12, Lemma 4]: does it assert #(Σ∩Z^d) ≤ C ρ^{k−1+ε} for every k-dimensional sphere of radius ρ with arbitrary center and affine span, with constant independent of all arithmetic parameters? If the published lemma has extra hypotheses, re-derive Lemma 6.2 with the actual factors; any additional R^c in (6.6) would invalidate (1.9). A numerical probe in d=4, k=2 with centers (a/q,0,0,0), q≍R, checking whether the count stays O(R^{1+ε}) uniformly in q, would provide evidence either way.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 2.1 is the single most load-bearing external input: (6.5) bounds #(E∩Σ_B) by R^η L_r, and the subsequent |X_B∩Z^d| bound uses the same lemma for spheres of dimension d−r−1. The relevant spheres have centers c_B that are circumcenters of lattice r-simplices; those denominators can grow like a power of R (the simplex determinant). The paper cites Huang–Zhang [12, Lemma 4] and says 'after relabeling the dimension', but it does not reproduce the lemma or prove that the constant is independent of center, determinant, covolume, or rationality. If the published lemma carries extra arithmetic hypotheses, or if the no-factor statement is false, the proof of Lemma 6.2 acquires an extra R^c factor; then the low-density endpoint estimate (1.9), and with it the sharp lower endpoint p=(d+2)/d in Theorem 1.1, would not follow. This is a correctness risk rather than a circularity: the rest of the proof is internally coherent, but this is the least secured step.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves fixed-radius ℓ^p-improving estimates for discrete spherical averages on Z^d. Theorem 1.1 asserts that for d≥4, with R^2∈N in dimensions d≥5 and R^2∈N∖4N in dimension 4, the probability average A_R satisfies ∥A_R f∥_{ℓ^{p'}} ≲ R^{-d(2/p-1)+ε} ∥f∥_{ℓ^p} for (d+2)/d ≤ p ≤ 2, uniformly in R. The proof separates into a circle-method regime above a density threshold and incidence-geometric regimes below it: the density threshold is handled by a refined low-frequency estimate plus an ℓ^2 high-frequency estimate, while the low-density regimes use rank-sensitive integer moments of the incidence function, an even-dimensional endpoint moment bound, an odd-dimensional rank-by-rank estimate, and, in four dimensions, a separate bridge estimate based on a point-sphere incidence result. The endpoint and exponent are shown to be sharp via box and point-mass examples. The stated theorem extends previous fixed-radius results of Kesler–Lacey and Hughes to the sharp lower endpoint.","tokens_in":17324,"tokens_out":6431,"duration_ms":70902,"significance":"If the result is correct, Theorem 1.1 resolves the sharp fixed-radius ℓ^p-improving range for discrete spherical averages in dimensions d≥4, and Corollary 1.3 gives the corresponding fixed-distance incidence bound. The proof is well structured: the local product density δ=|E||F|/R^{2d} is a natural parameter, the circle-method kernel is expressed through an exact Ramanujan-sum identity, and the sharpness examples are transparent. The paper also makes explicit use of machine-checkable decomposition steps and clearly identifies the new ingredient as the rank-sensitive integer-moment estimate. The publication value is high if the external inputs are verified.","major_comments":[{"comment":"The proof of the low-density endpoint rests on the uniform bound (2.1) for lattice points on arbitrary embedded spheres, with no dependence on center, determinant, covolume, or rationality. This lemma is applied in (6.5) to E∩Σ_B and again to X_B∩Z^d for spheres of dimension d−r−1 whose centers c_B are circumcenters of lattice r-simplices; the denominators of such circumcenters can grow with R. The manuscript cites Huang–Zhang [12, Lemma 4] 'after relabeling the dimension' but does not reproduce the lemma or verify that the claimed no-factor uniformity holds for these specific arithmetic positions. Since the moment estimate (6.6), Proposition 1.4, and the endpoint estimate (10.4) all depend on this uniformity, any extra R^c factor from Lemma 2.1 would break the sharp endpoint. This is load-bearing: please state and prove Lemma 2.1 in full, or quote the precise lemma with its hypotheses and confirm that the constant is independent of the center, determinant, covolume, and rationality.","section":"§2, Lemma 2.1; §6, (6.5) and Lemma 6.2"},{"comment":"Lemma 3.1 states the divisor sums (3.9)–(3.10) and says 'The proof below follows the decomposition of M2,2 and M2,3 in [18, Section 3]. We include the details, since the precise powers of q are needed in dimension four,' but no proof appears in the manuscript. These bounds are used in Proposition 3.2 to obtain the high-frequency ℓ^2 estimate, which in turn enters Propositions 4.2 and 5.1 and the circle-method density threshold. The statement cannot be verified as written, and the promised details are absent. Please supply the complete proof or a precise reference to the corresponding display in [18].","section":"§3.1, Lemma 3.1"},{"comment":"Theorem 8.1 is the only bridge for the four-dimensional low-density window R^2<|E|≤|F| and |E||F|≤C_1R^5, and it invokes 'Mudgal's four-dimensional incidence estimate' in [24] with only a coordinate hypothesis asserted. The statement and the full set of hypotheses of that external result are not reproduced, and the paper does not verify all assumptions (for example, any separation, boundedness, or general-position condition that the cited estimate may require). Since the conclusion (8.2) is essential to close d=4, please reproduce the theorem and check that the set E considered here satisfies every hypothesis.","section":"§8, Theorem 8.1"}],"minor_comments":[{"comment":"The average is denoted A_R in the theorem and displayed displays but frequently typeset as A R in the surrounding text; please standardize the notation.","section":"Throughout"},{"comment":"The lower endpoint is written as 'd+2/d' in two places instead of '(d+2)/d'; this is a typo that should be corrected.","section":"§11 and Section 1 statement of Theorem 1.1"},{"comment":"When d−r−1=0, the set X_B is described as a singleton and Lemma 2.1 is invoked with its O(1) convention. It would be useful to state explicitly in Lemma 2.1 how zero-dimensional spheres and radius-zero sections are counted, rather than leaving this to a parenthetical convention.","section":"§6.1, Lemma 6.1 and following"},{"comment":"The localization argument uses cubes Q_j of sidelength ⌈R⌉ and enlargements Q_j^*, but the exact properties required of Q_j^* (bounded overlap, |Q_j^*|≍R^d, and containment of all centers at distance R from Q_j) are only implicit. A sentence making these uniformities explicit would improve readability.","section":"§1, localization reduction"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this paper looks like it completes the sharp l^p-improving range for fixed-radius discrete spherical averages in all dimensions d>=4, including the d=4 class R^2 in N\\4N that Kesler-Lacey and Hughes left open. The main theorem is a genuine new result, not a repackaging: the endpoint p=(d+2)/d is strictly below previous ranges, and the proof introduces a rank-sensitive integer-moment incidence mechanism plus a four-dimensional bridge via Mudgal. The structure is clear, the local-to-global reduction is handled carefully, and Remark 1.2 correctly identifies the sharp exponent and endpoint. The paper is honest about its debts. The circle-method pieces adapt known mechanisms, and the references to the relevant prior work look right. The technical heart is the integer-moment estimate in Lemma 6.2 and the way it feeds into the low-density cases; that is the part where the paper is genuinely doing something new. Soft spots, in increasing order of worry. First, Lemma 3.1 is stated with a promise to prove it, but the proof never actually appears. The text says 'the proof below follows the decomposition...' and then moves to Proposition 3.2, which uses the lemma. That is a concrete omission and should be fixed. Second, and more seriously, the reliance on Lemma 2.1 is the real risk. That lemma, borrowed from Huang-Zhang, bounds lattice points on an arbitrary embedded sphere with no determinant, covolume, or rationality factor. The paper needs exactly that uniformity in Lemma 6.2, because the circumspheres of the lattice simplices have centers whose coordinates can accumulate denominators growing like a power of R. If Huang-Zhang's bound carries any hidden arithmetic restriction in this regime, the low-density endpoint collapses. The authors state the uniform version and cite [12, Lemma 4], but they do not reproduce the proof or verify the uniformity for the particular centers that arise here. That is the spot a referee should push hardest. I do not see circularity. The circle-method estimate and the incidence geometry are independent derivations, and the endpoint is not assumed in the proof. The two gaps above are enough to make me withhold final judgment, but not enough to dismiss the paper. This paper is for discrete harmonic analysts and anyone tracking l^p-improving estimates. It deserves a serious referee. My recommendation: send it to peer review, and ask the authors to actually prove Lemma 3.1 and to either reproduce the relevant part of Huang-Zhang's lemma or prove the uniform version they need. If those are satisfied, this is a clean, sharp result.","headline":"A genuinely sharp endpoint result for fixed-radius discrete spherical averages, with two concrete gaps to close: a missing proof of Lemma 3.1 and a black-box uniformity lemma that is load-bearing.","tokens_in":795,"tokens_out":1046,"would_cite":true,"duration_ms":31509,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B25","11P55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The fixed-radius discrete spherical average achieves its optimal improving range, with sharp decay and sharp endpoint, in every dimension at least four.","keywords":["discrete spherical averages","fixed-radius averages","ℓ^p-improving estimates","circle method","Ramanujan sums","discrete incidence bounds","restricted weak-type estimates","harmonic analysis on Z^d"],"falsifier":"A concrete check: for $d=5$, take $R\\to\\infty$ and finite sets $E_R,F_R$ inside a cube of side about $R$ with $|E_R||F_R|$ on the order of $R^{2d-\\kappa_d}$, and compare $\\langle A_R1_{E_R},1_{F_R}\\rangle/R^d$ with $R^{1/100}(|E_R||F_R|/R^{2d})^{d/(d+2)}$; exceeding the latter at infinitely many $R$ would disprove Proposition 1.4.","tokens_in":16921,"feed_emoji":"⚪","tokens_out":16502,"duration_ms":155196,"temperature":0.7,"pith_summary":"The paper studies the fixed-radius discrete spherical average $A_R f(x)=|S_R|^{-1}\\sum_{n\\in\\mathbb{Z}^d,\\,|n|=R}f(x-n)$ and proves that it maps $\\ell^p(\\mathbb{Z}^d)$ into $\\ell^{p'}(\\mathbb{Z}^d)$ with decay $R^{-d(2/p-1)+\\varepsilon}$ for the full range $(d+2)/d\\le p\\le 2$, whenever $d\\ge5$ and $R^2$ is any positive integer, or $d=4$ and $R^2$ is a positive integer not divisible by $4$. Both the decay exponent and the lower endpoint are sharp, so the paper settles the optimal fixed-radius $\\ell^p$-improving range. Earlier fixed-radius results either started later in $p$ or allowed constants that depend on the arithmetic of $R^2$; here the constant is uniform in $R$. The proof splits at a product-density threshold: above it, a circle-method estimate with Ramanujan-sum cancellation; below it, rank-sensitive integer-moment incidence counting on the lattice.","feed_headline":"Lattice-sphere averages are sharp across the full improving range","feed_subtitle":"In every dimension d≥4, the fixed-radius average reaches p=(d+2)/d with a constant uniform in R.","key_machinery":"The argument runs on two mechanisms. Above the density threshold $\\delta=|E||F|/R^{2d}\\ge R^{-\\kappa_d}$ with $\\kappa_d=(d-3)(d+2)/(d-2)$, a fixed-radius circle-method decomposition writes $A_R$ as a sum of low-frequency multipliers built from Ramanujan sums $c_q(|n|^2-R^2)$ plus a bounded high-frequency remainder; the exact-sphere contribution, where the Ramanujan argument vanishes, is absorbed into a small multiple of the original positive average. Below the threshold, the proof switches to geometric incidence counting. For the distance-incidence function $r_E(x)=\\#\\{e\\in E:|x-e|=R\\}$, the $j$-th integer moment counts configurations $(x,e_1,\\dots,e_j)$ with $|x-e_i|=R$. These are organized by the affine rank $r$ of $\\{e_1,\\dots,e_j\\}$: for a fixed affine basis $B$, the remaining points lie on the circumsphere $\\Sigma_B$, while the admissible centers $x$ lie on the orthogonal embedded sphere $X_B$. Lemma 6.2 bounds each rank-$r$ contribution using the uniform lattice-point estimate for embedded spheres, $\\#(\\Sigma\\cap\\mathbb{Z}^d)\\lesssim_{d,\\varepsilon}\\rho^{k-1+\\varepsilon}$, and the theorem closes by matching the resulting moment bounds to the endpoint exponent in even, odd, and four-dimensional cases.","core_discovery":"The central claim is Theorem 1.1: for every $\\varepsilon>0$, $\\|A_R f\\|_{\\ell^{p'}(\\mathbb{Z}^d)}\\lesssim_{d,p,\\varepsilon} R^{-d(2/p-1)+\\varepsilon}\\|f\\|_{\\ell^p(\\mathbb{Z}^d)}$ for $(d+2)/d\\le p\\le2$, under the stated arithmetic condition on $R^2$. The power of $R$ cannot be increased uniformly, because a large indicator box already has $\\|A_R\\|_{\\ell^p\\to\\ell^{p'}}\\gtrsim R^{-d(2/p-1)}$, and the range cannot be extended below $p=(d+2)/d$, because a point mass gives $\\|A_R\\delta_0\\|_{\\ell^{p'}}=|S_R|^{-1/p}\\asymp R^{-(d-2)/p}$, which outgrows the estimate exactly when $p<(d+2)/d$. In dimension four, the exclusion of $R^2\\in4\\mathbb{N}$ is necessary: for $R=2^j$ the sphere has only $24$ points and no decay is possible. The theorem is proved by first establishing the localized restricted weak-type endpoint at $p=(d+2)/d$, then interpolating with the trivial $\\ell^1\\to\\ell^\\infty$ and $\\ell^2\\to\\ell^2$ bounds.","pith_inferences":["The uniformity in $R$ is the kind of ingredient a sparse-maximal or full-maximal theorem for fixed-radius averages would need; if the same bounds survive a summation over radii, the corresponding maximal operator should inherit the sharp improving range, though the paper does not address this.","The rank-sensitive moment strategy is not tied to spheres: adapting Lemma 6.2 to lattice points on other algebraic hypersurfaces, for instance discrete paraboloids or moment curves, could lower the known improving endpoint in those problems.","The proof leaves an $R^\\varepsilon$ loss at the endpoint while the sharp examples force only a slower logarithmic-type obstruction; eliminating the loss would require a divisor-sum improvement of Lemma 3.5 rather than a change in the geometric counting."],"forward_implications":["The fixed-distance incidence estimate $I_R(E,F)\\lesssim_{d,\\varepsilon} R^{2(d-2)/(d+2)+\\varepsilon}(|E||F|)^{d/(d+2)}$ holds for all finite $E,F\\subset\\mathbb{Z}^d$, with the same uniformity in $R$; this is Corollary 1.3.","Popular centers are controlled: if $r_E(x)\\ge t$, then the number of such $x$ is at most $O_{d,\\varepsilon}(R^{d-2+\\varepsilon}|E|^{d/2}t^{-(d+2)/2})$.","At the endpoint, the restricted weak-type norm satisfies $\\|A_R\\|_{\\ell^{(d+2)/d,1}\\to\\ell^{(d+2)/2,\\infty}}\\lesssim_{d,\\varepsilon} R^{-d(d-2)/(d+2)+\\varepsilon}$.","The estimate holds with constants uniform over all admissible $R^2$, removing any dependence on the number of prime factors of $R^2$ that appeared in earlier fixed-radius circle-method arguments.","Sharpness is intrinsic: the large-box example fixes the decay exponent and the point-mass example fixes the range, so no further endpoint improvement is possible without changing the normalization of the operator."],"supporting_citations":[{"why":"Gives the classical formula for the number of representations as a sum of five or more squares, yielding $|S_R|\\asymp R^{d-2}$ for $d\\ge5$.","marker":"[10]"},{"why":"Gives the four-squares formula, yielding the lower bound $|S_R|\\ge8R^2$ used in the dimension-four normalization.","marker":"[11]"},{"why":"Supplies the uniform lattice-point bound for embedded spheres, Lemma 2.1, used throughout the rank-sensitive moment estimate.","marker":"[12]"},{"why":"Supplies the fixed-cutoff circle-method decomposition into low- and high-frequency multipliers and the residual $\\ell^2$ bound.","marker":"[13]"},{"why":"Supplies the divisor-sum estimates and explicit exponential-sum bounds used in the high-frequency $\\ell^2$ estimate.","marker":"[18]"},{"why":"Supplies the zero-parameter separation mechanism: cancellation away from the exact sphere and absorption of the exact-sphere contribution.","marker":"[20]"},{"why":"Provides the four-dimensional point-sphere incidence estimate that bridges the remaining low-density window in $d=4$.","marker":"[24]"}],"fun_headline_variants":["Lattice-sphere averages hit sharp p-improving bound","Fixed-radius lattice spheres: sharp endpoint achieved","Sharp lattice-sphere estimates across full improving range","Lattice-sphere averages: sharp decay for fixed R","Optimal p-improving for discrete spherical averages"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole low-density half of the proof rests on the uniform bound that every nondegenerate $k$-dimensional sphere embedded in $\\mathbb{R}^d$, no matter where its center sits and whether its parameters are rational or not, contains at most $O_\\varepsilon(\\rho^{k-1+\\varepsilon})$ lattice points.","fun_headline_variants_meta":{"raw":{"variants":["Lattice-sphere averages hit sharp p-improving bound","Fixed-radius lattice spheres: sharp endpoint achieved","Sharp lattice-sphere estimates across full improving range","Lattice-sphere averages: sharp decay for fixed R","Optimal p-improving for discrete spherical averages"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000323,"raw_usage":{"total_tokens":1846,"prompt_tokens":1010,"completion_tokens":836,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":626,"completion_tokens_details":{"reasoning_tokens":762}},"tokens_in":626,"tokens_out":836,"duration_ms":8541,"temperature":1.0,"reasoning_tokens":762,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:23:56.744757+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check: for $d=5$, take $R\\to\\infty$ and finite sets $E_R,F_R$ inside a cube of side about $R$ with $|E_R||F_R|$ on the order of $R^{2d-\\kappa_d}$, and compare $\\langle A_R1_{E_R},1_{F_R}\\rangle/R^d$ with $R^{1/100}(|E_R||F_R|/R^{2d})^{d/(d+2)}$; exceeding the latter at infinitely many $R$ would disprove Proposition 1.4.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the classical formula for the number of representations as a sum of five or more squares, yielding $|S_R|\\asymp R^{d-2}$ for $d\\ge5$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the four-squares formula, yielding the lower bound $|S_R|\\ge8R^2$ used in the dimension-four normalization."},{"cited_title":"Restriction of toral eigenfunctions to totally geodesic submanifolds","cited_arxiv_id":"1902.09019","evidence_quote":"Supplies the uniform lattice-point bound for embedded spheres, Lemma 2.1, used throughout the rank-sensitive moment estimate."},{"cited_title":"The discrete spherical averages over a family of sparse sequences","cited_arxiv_id":"1609.04313","evidence_quote":"Supplies the fixed-cutoff circle-method decomposition into low- and high-frequency multipliers and the residual $\\ell^2$ bound."},{"cited_title":"$\\ell^p$-improving inequalities for Discrete Spherical Averages","cited_arxiv_id":"1804.09845","evidence_quote":"Supplies the divisor-sum estimates and explicit exponential-sum bounds used in the high-frequency $\\ell^2$ estimate."},{"cited_title":"Sparse Bounds for the Discrete Spherical Maximal Function","cited_arxiv_id":"1810.02240","evidence_quote":"Supplies the zero-parameter separation mechanism: cancellation away from the exact sphere and absorption of the exact-sphere contribution."},{"cited_title":"Additive energies on spheres","cited_arxiv_id":"2105.06925","evidence_quote":"Provides the four-dimensional point-sphere incidence estimate that bridges the remaining low-density window in $d=4$."}],"review_version":1}