{"id":"f7449679-d5f8-4bea-9260-c94e479c3d8a","arxiv_id":"2505.24699","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Geometric Littlewood-Offord probabilities are bounded by counting lattice points, resolving conjectures for varieties, convex-position sets, and bounded Chow-rank polynomials.","lead":"Random signed sums of vectors rarely land inside algebraic curves, convex shells, or other special sets; this paper proves sharp bounds by counting lattice points on those sets. It settles two open conjectures in geometric Littlewood-Offord theory and confirms a conjecture for low-complexity polynomials.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader's verdict ACCEPT is supported by the detailed and coherent proof structure. The reader identified Theorem 6.8 as the weakest assumption, but that theorem is load-bearing only for the secondary polynomial result Theorem 1.1(2); the central Theorem 1.3 rests on Pila's Theorem 6.7 and the affine-subspace case. I agree that there is no critical flaw or unmet hypothesis in the central argument. I mark agreement as partial because the reader's singled-out assumption is not the one most directly supporting the central claim, though both are deep imported number-theoretic inputs. Since I found no significant objection, no verdict adjustment is needed.","tokens_in":31381,"tokens_out":56087,"duration_ms":699322,"concrete_test":"Recompute the final step of Theorem 1.4 by substituting Pila's density bound d_S(B) = O(B^{-(k-l+1-1/d)}) and the Schwartz-Zippel term (b log b)^{-(k-l+1)/2} into Theorem 8.1, verifying that the second term is dominated for all 0 <= l <= k and d >= 2. Also verify that the constant alpha = (2k(2k)^k)^{-1} from Theorem 7.1 guarantees m' >= alpha b k, so Proposition 3.8 applies with a factor depending only on k.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the central claim (Theorem 1.3) as a corollary of Theorem 1.4 via Theorem 8.1, together with the degree-1 affine-subspace case handled by Ferber–Jain–Zhao. The iterative decoupling argument in Lemma 7.2, Proposition 7.4, and Theorem 7.1 is internally consistent: the subspace U strictly decreases in dimension, the basis-packing loss is tracked through the constants, and the procedure terminates in at most k steps. The reduction to lattice-point density in Theorem 4.1 is also sound: Fact 3.2 is used in the correct direction for subsequences, Proposition 3.7 correctly identifies densities under preimages, and the union bound over lattice points in the dilated GAP is legitimate. The deepest external input actually load-bearing for Theorem 1.3 is Pila's Theorem 6.7, which is used for arbitrary irreducible complex varieties under affine transformations. This is a standard uniform determinant-method bound, and the paper's use is consistent with the cited statements. The d != 3 restriction in Theorem 6.8 affects only Theorem 1.1(2), not the central Kwan–Sauermann conjecture. I found no load-bearing flaw in the proof of the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a general method for bounding the probability that a Rademacher-weighted sum of vectors from a sequence A falls into a target set S, by reducing the problem to counting lattice points in affine images of S. The central result is Theorem 1.3, which proves the Kwan–Sauermann conjecture: if the vectors a_1,...,a_n in C^k contain b disjoint bases and S is an affine algebraic variety of dimension at most l and degree at most d, then the probability is O_{d,k}(b^{-(k-l)/2}). The paper also proves a refined estimate (Theorem 1.4) for irreducible varieties using Pila's lattice-point bound, and a general transfer theorem (Theorem 8.1). Applications include: resolution of the Fox–Kwan–Spink conjecture for sets in convex position (Theorems 1.6 and 1.7), removal of logarithmic factors for semialgebraic sets containing no line segment (Theorem 1.8), and new results in the polynomial Littlewood–Offord problem for polynomials of bounded Chow rank (Theorems 1.1 and 1.2), including confirmation of the Nguyen–Vu conjecture in this special case and a repaired Costello-type bound up to the d=3 exception inherited from the affine dimension growth conjecture. The proof architecture consists of an optimal inverse Littlewood–Offord theorem, a new decoupling decomposition of the ambient space into 'structured' and 'disordered' subspaces (Theorem 7.1), and lattice-point density estimates.","tokens_in":31579,"tokens_out":47021,"duration_ms":427790,"significance":"If valid, this is a significant contribution to Littlewood–Offord theory. The resolution of the Kwan–Sauermann conjecture settles the natural geometric generalization of the classical linear problem, and the general framework connecting anticoncentration to lattice-point counting is likely to be influential. The authors are careful to isolate the d=3 case where the external dimension-growth input is open, and the internal reductions cancel cleanly: in particular, the inverse theorem's rho(A')^{-1} factor is canceled by the lattice-point union bound, and the iterative decoupling terminates by dimension. The paper also gives sharp (up to logarithmic factors) bounds for polynomials of bounded Chow rank, confirming a conjecture of Nguyen and Vu in that setting. The proofs are detailed and the main steps are checkable; I found no load-bearing error in the central derivation.","major_comments":[],"minor_comments":[{"comment":"The proof applies Theorem 4.4 with s1 = delta*n/2 but does not specify the epsilon needed for the hypothesis n^epsilon <= s1; one can fix any epsilon in (0,1) (e.g., epsilon=1/2) and argue for sufficiently large n, with the small-n case absorbed into the O-constant. The same remark applies to the use of Theorem 4.3 in Proposition 7.4. Please add a sentence clarifying this.","section":"Section 4, proof of Theorem 4.1"},{"comment":"The statement 'F^=_* - F has at most (b/2+d)n^{d-1} nonzero coefficients' is not literally meaningful because F^=_* is a polynomial in the uneliminated variables only; the intended comparison is with the n-variable polynomial obtained by extending F^=_* with zero coefficients on the eliminated variables. Please rephrase.","section":"Section 8, proof of Theorem 1.1(2)"},{"comment":"The notation 'F =d_*' for the degree-d homogeneous part of F_* is confusing; I suggest a standard notation such as F_*^{(d)}.","section":"Section 8, proof of Theorem 1.1(2)"},{"comment":"For the application to the density function d_S(B), it would be helpful to note explicitly that the hypothesis 'f cannot be represented as a polynomial of two linear forms' is preserved under invertible affine-linear changes of variables, so the bound N_{phi(S)}(B) is uniform over the affine transformations phi used in Definition 3.6.","section":"Section 6.2, after Theorem 6.8"},{"comment":"The proof states 'By decreasing s1, we may assume s1 <= n/2' and then applies Theorem 4.3 to subsequences of size n_i; it may be worth noting that the constant C in rho(A_i) >= n_i^{-C} remains uniform because n_i >= n/2 throughout the iteration.","section":"Section 4, proof of Theorem 4.4"}],"recommendation":"minor_revision","confidential_remarks":"This is a well-written and mathematically substantial paper. I found no load-bearing errors; the minor issues are presentation and technical clarity points. I recommend minor revision and expect acceptance after these local fixes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is one of the strongest Littlewood–Offord papers in a while. The main novelty is the reduction in Theorems 4.1 and 8.1: for sequences of vectors with good basis-packing, the probability of hitting a variety S is controlled by the affine-invariant lattice-point density d_S. That is a clean, reusable idea, and it is exactly what lets them resolve the Fox–Kwan–Spink conjecture for convex position (Theorem 1.6) and the Kwan–Sauermann conjecture for algebraic varieties (Theorem 1.3). The corollaries for bounded Chow-rank polynomials (Theorems 1.1 and 1.2) are genuinely new, including the Nguyen–Vu confirmation and the progress toward the repaired Costello conjecture.\n\nI read the central proof chain and found it coherent. Theorem 4.1 iterates the Nguyen–Vu inverse Littlewood–Offord theorem, and the density function's invariance under preimages (Proposition 3.7) is proved correctly and does the work. The decoupling in Lemma 7.2 is consistent: the variety dimension drops in the dim T <= l-1 case, and the dim T = l case forces a symmetry subspace V_S whose probability is controlled by rho(A0, V_S). The iteration in Theorem 7.1 terminates because U shrinks. The stress-test note is right: there is no load-bearing flaw.\n\nThe soft spots are real but proportionate. The d=3 gap in Theorem 1.1(2) is inherited from the affine dimension-growth conjecture for hypersurfaces; it is stated openly and only affects the Costello-type b^{-1+eps} bound, not the main Kwan–Sauermann theorem. The paper relies on several deep external inputs (Pila's determinant method, Vermeulen/Browning–Gorodnik, Nguyen–Vu), and the proofs are long—nobody has machine-checked them. That is normal for this area, but it justifies moderate rather than high confidence. There is some self-citation (Kwan coauthored [16] and [25]), but the conjectures are resolved using different machinery; the reasoning is not circular.\n\nThis paper is for researchers in additive combinatorics, anticoncentration, and random polynomials, and it deserves a serious referee. I would send it to a strong combinatorialist and a number theorist. It should be accepted and will be well cited.","headline":"Resolves two named conjectures with a genuinely reusable reduction; the long proofs look sound, with the main weakness being an honestly flagged d=3 gap inherited from number theory.","tokens_in":32132,"tokens_out":2061,"would_cite":true,"duration_ms":25286,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11P21","11D45","14G05","60E15","52B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that random signed sums of vectors containing $b$ disjoint bases hit an algebraic variety of dimension at most $\\ell$ and degree at most $d$ with probability at most $O_{d,k}(b^{-(k-\\ell)/2})$, via a reduction to counting…","keywords":["Littlewood–Offord problem","Rademacher sums","algebraic varieties","lattice point counting","inverse Littlewood–Offord theorem","Chow rank","convex position","anticoncentration"],"falsifier":"Compute the hitting probability in the sharpness example behind Example 1.5: take $2m$ copies of $e_i/2$ in each coordinate, set $b=2m$, and choose $S=\\{x_1=0\\}$; the probability is $\\binom{2m}{m}/2^{2m}=\\Theta(b^{-1/2})$, so any sequence of $b$ disjoint bases with hitting probability $\\omega(b^{-(k-\\ell)/2})$ on a degree-$d$ variety of dimension $\\ell$ would refute Theorem 1.3. For the degree-3 gap, an explicit irreducible cubic hypersurface not of two linear forms with more than $C_\\varepsilon B^{k-2+\\varepsilon}$ integer points in $[-B,B]^k$ would show why the $b^{-1+\\varepsilon}$ argument cannot currently extend to degree 3.","tokens_in":31170,"feed_emoji":"🎲","tokens_out":14740,"duration_ms":126827,"temperature":0.7,"pith_summary":"The paper proves that geometric Littlewood–Offord probabilities can be read off from lattice-point counts. Its central result is that if vectors $a_1,\\dots,a_n\\in\\mathbb{C}^k$ contain $b$ disjoint bases and $S\\subset\\mathbb{C}^k$ is an algebraic variety of dimension at most $\\ell$ and degree at most $d$, then $\\mathbb{P}[\\xi_1a_1+\\cdots+\\xi_na_n\\in S]\\le O_{d,k}(b^{-(k-\\ell)/2})$, resolving the Kwan–Sauermann conjecture. The same method gives an asymptotically sharp $(2\\sqrt{2/\\pi}+o(1))n^{-1/2}$ bound for hitting sets in convex position, an $O_S(n^{-1/2})$ bound for semialgebraic sets with no line segment, and polynomial Littlewood–Offord bounds for polynomials of bounded Chow rank, including the Nguyen–Vu conjecture in that setting and a near-$b^{-1+\\varepsilon}$ bound under robust irreducibility for degree $d\\ne 3$. The proof reduces the problem to counting integer points in affine images of $S$: a concentrated random sum is shown to be nearly uniform over a box in a lattice, and an iterative decomposition handles sequences that are not initially concentrated.","feed_headline":"Signed sums hit algebraic sets with sharp power-law odds","feed_subtitle":"Counting lattice points on the target set resolves the Kwan–Sauermann conjecture and sharpens polynomial Littlewood–Offord bounds.","key_machinery":"The load-bearing object is the lattice-point density $d_S(B)=\\sup_{\\varphi}N_{\\varphi(S)}(B)/(2\\lfloor B\\rfloor+1)^k$, maximized over bijective affine-linear maps $\\varphi$. Theorem 4.1 shows that in the concentrated regime, where the maximum point probability is at least $n^{-C}$, the translate probability $\\rho(A,S)$ is at most $d_S(\\sqrt{n\\log n})(\\log n)^r$ plus a negligible term; the proof iterates the optimal inverse Littlewood–Offord theorem to trap almost all coefficient vectors in a proper symmetric generalized arithmetic progression of bounded rank, transfers the sum to an integer box through the progression's generators, and applies Hoeffding's inequality together with the definition of $d_S$. Theorem 7.1 extends this to arbitrary sequences by an iterative decoupling argument that splits $\\mathbb{C}^k$ into a structured subspace $W$, where the projected sum is polynomially concentrated, and a disordered subspace $U$, where the uncontrolled part of $S$ has negligible probability. Number-theoretic input enters only through $d_S$: Schwartz–Zippel for the baseline count, Pila's $B^{\\ell-1+1/d}(\\log B)^C$ bound for irreducible varieties, and the dimension-growth estimate $B^{k-2+\\varepsilon}$ for hypersurfaces of degree $d\\ne 3$.","core_discovery":"On the paper's own terms, the central discovery is that the extremal quantity is the affine-invariant lattice-point density of the target set, not its shape. Theorem 1.3 states the sharp bound above for all algebraic varieties, possibly reducible, and Theorem 1.4 refines it for irreducible $S$ to $O_{d,k}(b^{-(k-\\ell+1-1/d)/2}(\\log b)^{C_{d,k}})$. These are deduced from Theorem 8.1, which bounds the maximum translate probability $\\rho(A,S)$ by $\\bigl(d_S(\\sqrt{b\\log b})+(b\\log b)^{-(k-\\ell+1)/2}\\bigr)(\\log b)^r$, where $d_S(B)$ is the supremum over affine-linear changes of coordinates of the proportion of integer points in $[-B,B]^k$ lying in $S$. Using Pila's determinant-method estimate for irreducible varieties gives the refined exponent; using the affine dimension-growth estimates of Vermeulen and Browning–Gorodnik for hypersurfaces of degree $d\\ne 3$ gives the $b^{-1+\\varepsilon}$ polynomial bound. The paper explicitly identifies the degree-3 exception as a gap in the available uniform estimates rather than a limitation of the method itself.","pith_inferences":["The lattice-point reduction is modular: a proof of the affine dimension-growth conjecture for degree 3 would automatically transfer the $b^{-1+\\varepsilon}$ bound to degree 3 without touching the probabilistic arguments.","The same black-box structure suggests a broader principle: any class of sets whose affine images have sufficiently strong integer-point bounds should inherit corresponding Littlewood–Offord estimates, so one could test the method on sets definable in o-minimal structures rather than only semialgebraic ones.","An unexplored stress test is to replace Chow rank by Schmidt or partition rank in the polynomial corollaries; the decoupling decomposition may survive, but the lattice-point density step would need a new estimate adapted to low-rank coordinates."],"forward_implications":["Theorem 1.3 settles the Kwan–Sauermann conjecture with the expected exponent $(k-\\ell)/2$ in full generality, covering reducible varieties.","Theorem 1.1(1) gives $O_{d,c}(b^{-1/2})$ for bounded-Chow-rank polynomials that robustly depend on $b$ variables, matching the Nguyen–Vu conjecture in this case and best possible up to constants.","Theorem 1.1(2) gives $O_{d,c,\\varepsilon}(b^{-1+\\varepsilon})$ for robustly irreducible bounded-Chow-rank polynomials of degree $d\\ne 3$, a step toward the repaired Costello conjecture.","Theorems 1.6 and 1.8 show that for convex-position sets and for semialgebraic sets without line segments, no robust spanning assumption is needed to obtain $O(n^{-1/2})$ type bounds.","Theorem 1.2 yields $b^{-1+1/(2d)}(\\log b)^{C_{d,c}}$ for polynomials irreducible over a subfield of $\\mathbb{C}$, even when they factor over $\\mathbb{C}$."],"supporting_citations":[{"why":"introduced the convex-position and o-minimal Littlewood–Offord questions; its Theorem 1.9(2) supplies the spread-out case in the proof of Theorem 1.7.","marker":"[16]"},{"why":"raised the conjecture that Theorem 1.3 resolves and proved its quadratic case; its robust-dependence assumption is used in Theorem 1.1(1).","marker":"[25]"},{"why":"the optimal inverse Littlewood–Offord theorem used to build the bounded-rank generalized arithmetic progression in Theorem 4.4 and Proposition 7.4.","marker":"[28]"},{"why":"Pila's determinant-method bound on integer points in irreducible varieties gives the refined exponent in Theorem 1.4.","marker":"[30, 31]"},{"why":"Vermeulen's affine dimension-growth estimate for hypersurfaces of degree at least 4 supplies the $b^{-1+\\varepsilon}$ bound in Theorem 1.1(2).","marker":"[37]"},{"why":"Browning–Gorodnik's affine dimension-growth estimate for degree 2 completes the $d\\ne 3$ case of Theorem 1.1(2).","marker":"[5]"},{"why":"Andrews' bound on lattice vertices of convex bodies controls the lattice-point density $d_S$ in the proof of Theorem 1.7.","marker":"[1]"},{"why":"the refined Halász theorem controls affine-subspace hitting probabilities in the degree-1 case of Theorem 1.3.","marker":"[15]"},{"why":"the Erdős–Littlewood–Offord theorem supplies the one-dimensional case in the deduction of Theorem 1.6.","marker":"[14]"}],"fun_headline_variants":["Lattice point census solves geometric Littlewood-Offord","Sharp signed-sum hit odds from lattice density","Conjectures resolved: lattice counting for random sums","Algebraic sets resist signed sums via lattice point limits","Geometric Littlewood-Offord tamed by affine lattice counts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the affine dimension-growth estimate for hypersurfaces of degree $d\\ne 3$ — that an irreducible polynomial not expressible through two linear forms has $O_{d,k,\\varepsilon}(B^{k-2+\\varepsilon})$ integer zeros in a box — since Theorem 1.1(2) collapses if this estimate fails or the degree-3 case is needed.","fun_headline_variants_meta":{"raw":{"variants":["Lattice point census solves geometric Littlewood-Offord","Sharp signed-sum hit odds from lattice density","Conjectures resolved: lattice counting for random sums","Algebraic sets resist signed sums via lattice point limits","Geometric Littlewood-Offord tamed by affine lattice counts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00024,"raw_usage":{"total_tokens":1602,"prompt_tokens":1115,"completion_tokens":487,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":731,"completion_tokens_details":{"reasoning_tokens":409}},"tokens_in":731,"tokens_out":487,"duration_ms":5322,"temperature":1.0,"reasoning_tokens":409,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:20:37.048788+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the hitting probability in the sharpness example behind Example 1.5: take $2m$ copies of $e_i/2$ in each coordinate, set $b=2m$, and choose $S=\\{x_1=0\\}$; the probability is $\\binom{2m}{m}/2^{2m}=\\Theta(b^{-1/2})$, so any sequence of $b$ disjoint bases with hitting probability $\\omega(b^{-(k-\\ell)/2})$ on a degree-$d$ variety of dimension $\\ell$ would refute Theorem 1.3. For the degree-3 gap, an explicit irreducible cubic hypersurface not of two linear forms with more than $C_\\varepsilon B^{k-2+\\varepsilon}$ integer points in $[-B,B]^k$ would show why the $b^{-1+\\varepsilon}$ argument cannot currently extend to degree 3.","supporting_citations":[{"cited_title":"Geometric and o-minimal Littlewood-Offord problems","cited_arxiv_id":null,"evidence_quote":"introduced the convex-position and o-minimal Littlewood–Offord questions; its Theorem 1.9(2) supplies the spread-out case in the proof of Theorem 1.7."},{"cited_title":"Resolution of the quadratic Littlewood–Offord problem","cited_arxiv_id":null,"evidence_quote":"raised the conjecture that Theorem 1.3 resolves and proved its quadratic case; its robust-dependence assumption is used in Theorem 1.1(1)."},{"cited_title":"Optimal inverse Littlewood-Offord theorems","cited_arxiv_id":null,"evidence_quote":"the optimal inverse Littlewood–Offord theorem used to build the bounded-rank generalized arithmetic progression in Theorem 4.4 and Proposition 7.4."},{"cited_title":"Dimension growth for affine varieties","cited_arxiv_id":null,"evidence_quote":"Vermeulen's affine dimension-growth estimate for hypersurfaces of degree at least 4 supplies the $b^{-1+\\varepsilon}$ bound in Theorem 1.1(2)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Browning–Gorodnik's affine dimension-growth estimate for degree 2 completes the $d\\ne 3$ case of Theorem 1.1(2)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Andrews' bound on lattice vertices of convex bodies controls the lattice-point density $d_S$ in the proof of Theorem 1.7."},{"cited_title":"On the number of Hadamard matrices via anti-concentration","cited_arxiv_id":null,"evidence_quote":"the refined Halász theorem controls affine-subspace hitting probabilities in the degree-1 case of Theorem 1.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the Erdős–Littlewood–Offord theorem supplies the one-dimensional case in the deduction of Theorem 1.6."}],"review_version":1}