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Geometric Littlewood-Offord problems via lattice point counting

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2505.24699 v1 pith:G2TBRIYG submitted 2025-05-30 math.CO math.NTmath.PR

Alexandr Grebennikov , Matthew Kwan This is my paper
classification math.COmath.NTmath.PR MSC 11P2111D4514G0560E1552B20
keywords Littlewood–OffordproblemRademachersumsalgebraicvarietieslatticepointcountinginversetheoremChowrankconvexpositionanticoncentration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $d\ne 3$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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 $\mathbb{C}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

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.

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.

minor comments (5)
  1. [Section 4, proof of Theorem 4.1] 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.
  2. [Section 8, proof of Theorem 1.1(2)] 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.
  3. [Section 8, proof of Theorem 1.1(2)] The notation 'F =d_*' for the degree-d homogeneous part of F_* is confusing; I suggest a standard notation such as F_*^{(d)}.
  4. [Section 6.2, after Theorem 6.8] 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.
  5. [Section 4, proof of Theorem 4.4] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivation is self-contained once external lattice-point and inverse Littlewood-Offord theorems are granted.

full rationale

The central claim (Theorem 1.3) is derived from Theorem 1.4, which is proved from Theorem 8.1 combined with Pila's Theorem 6.7. Theorem 8.1 is an honest reduction to the lattice-point density d_S: its proof splits into the structured case via Theorem 7.1 and Theorem 4.1, and the complementary case via Bezout and Schwartz-Zippel, with constants tracked throughout. The decomposition in Theorem 7.1 is iterative and terminates because the subspace U strictly decreases in dimension; its one-step Proposition 7.4 uses the optimal inverse Littlewood-Offord theorem of Nguyen-Vu, an external result, together with the decoupling Lemma 7.2, which is proved in the paper. No step assumes the Kwan-Sauermann or Fox-Kwan-Spink conjectures that the paper claims to resolve. The only co-authored citation, [16, Theorem 1.9(2)] (Fox-Kwan-Spink, a paper coauthored by Kwan), is used for the convex-position result Theorem 1.7 and is an independently published external theorem with its own proof; it is not the basis of Theorem 1.3 or of the algebraic main results. Pila's theorem and the dimension-growth estimates are external number-theoretic inputs whose stated assumptions do not include the target probability bounds. The d != 3 restriction in Theorem 6.8 is explicitly disclosed and affects only Theorem 1.1(2), not the central Theorem 1.3. I therefore find no circular step.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new entities. It rests on published theorems from additive combinatorics, diophantine geometry, and discrete geometry; the only author-overlap input is the Fox-Kwan-Spink spread-out bound, which is independent of the conjectures being resolved.

assumptions (6)
  • standard math Optimal inverse Littlewood-Offord theorem (Nguyen-Vu): if rho(A) >= n^{-C}, then all but s elements of A lie in a proper symmetric GAP of rank O(1) and volume O(rho^{-1} s^{-r/2}).
    Used as Theorem 4.3 and iterated in Theorem 4.4 to drive Theorem 4.1, the paper's core reduction to lattice-point counting.
  • standard math Affine dimension growth for hypersurfaces of degree d != 3 (Vermeulen; Browning-Gorodnik): irreducible f not a polynomial of two linear forms satisfies N_S(B) <= O_{d,k,eps}(B^{k-2+eps}).
    Used in Section 8 to bound d_S in the proof of Theorem 1.1(2); the d=3 case remains open and is excluded from the theorem.
  • standard math Pila's lattice-point estimate for irreducible varieties: N_S(B) <= O_{d,k}(B^{l-1+1/d}(log B)^{C_d}).
    Used in Theorem 1.4 and Theorem 1.2 through Theorem 8.1 to control the density of integer points on irreducible varieties.
  • standard math Fox-Kwan-Spink spread-out bound: for convex-position S, rho(A,S) <= O_k(rho(A)^{1/(k 2^{k-1})}).
    Theorem 5.2 handles the spread-out case of Theorem 1.7; it comes from a paper coauthored by Kwan but is a distinct prior result.
  • standard math Andrews' bound on lattice points in convex position: N_S(B) <= O_k(B^{k-2k/(k+1)}).
    Used to bound the lattice-point density in the convex-position case, producing the exponent in Theorem 1.7.
  • standard math Ferber-Jain-Zhao refined Halasz theorem for point probabilities of sums of vectors partitioned into spanning blocks.
    Used to handle degree-1 affine subspace components in Theorem 1.3.

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Pith. "Pith review of Geometric Littlewood-Offord problems via lattice point counting." pith.science (2026). https://pith.science/paper/G2TBRIYG

@misc{pith2026250524699,
  author       = {Pith},
  title        = {Pith review of: Geometric Littlewood-Offord problems via lattice point counting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G2TBRIYG}},
  note         = {Machine review of arXiv:2505.24699}
}
abstract

Consider nonzero vectors $a_{1},\dots,a_{n}\in\mathbb{C}^{k}$, independent Rademacher random variables $\xi_{1},\dots,\xi_{n}$, and a set $S\subseteq\mathbb{C}^{k}$. What upper bounds can we prove on the probability that the random sum $\xi_{1}a_{1}+\dots+\xi_{n}a_{n}$ lies in $S$? We develop a general framework that allows us to reduce problems of this type to counting lattice points in $S$. We apply this framework with known results from diophantine geometry to prove various bounds when $S$ is a set of points in convex position, an algebraic variety, or a semialgebraic set. In particular, this resolves conjectures of Fox-Kwan-Spink and Kwan-Sauermann. We also obtain some corollaries for the polynomial Littlewood-Offord problem, for polynomials that have bounded Chow rank (i.e., can be written as a polynomial of a bounded number of linear forms). For example, one of our results confirms a conjecture of Nguyen and Vu in the special case of polynomials with bounded Chow rank: if a bounded-degree polynomial $F\in\mathbb{C}[x_{1},\dots,x_{n}]$ has bounded Chow rank and ''robustly depends on at least $b$ of its variables'', then $\mathbb{P}[F(\xi_{1},\dots,\xi_{n})=0]\le O(1/\sqrt{b})$. We also prove significantly stronger bounds when $F$ is ''robustly irreducible'', towards a conjecture of Costello.

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