Hollom sorkin 2025 reverse littlewood offord parity conditions

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evidence/: Holds the independent review record of the repaired higher-dimensional chain for Theorem 1.5 and the reviewed text; no executable evidence is filed here.

repaired_higher_dimensional_chain: Compilation-supplied reconstruction of the proof of Theorem 1.5 for d at least 3 with the paper's constant epsilon = 2^{-100} d^{-80}, correcting the twelve printed issues; author-recorded, not independently accepted.

theorem_1_3: Paired planar unit vectors at angles arcsin of powers of 1/20, plus (1,0), have a Rademacher signed sum in the closed unit disk with probability exactly 2 to the minus floor of n/2; recorded at statement depth.

theorem_1_5: For unit vectors in d-space with n of parity opposite to d, some signing has norm at most root of d minus epsilon, with epsilon = 2^{-100} d^{-80}; the printed proof for d at least 3 is incomplete and a repaired chain is recorded separately.

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Lawrence Hollom and Gregory B. Sorkin, Reverse Littlewood–Offord problems with parity conditions. arXiv:2510.05044v1 [math.CO], 6 October 2025, 13 pages.

Source identity and local artifact

The selected PDF is the arXiv v1 manuscript (watermark "arXiv:2510.05044v1 [math.CO] 6 Oct 2025" on p. 1), 13 physical pages whose printed numbers equal the PDF page numbers. Provenance: downloaded from https://arxiv.org/abs/2510.05044 on 2026-09-05; 387,239 bytes. The survey download set of September 2026 records the author listing as pointing to this arXiv record with no published replacement; no later version was acquired. The arXiv record (https://arxiv.org/abs/2510.05044, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Read status: claims checked for Theorems 1.3 and 1.5 on the page images; all 13 pages were read at filing. The proof of Theorem 1.3 (Section 3, p. 5) was read. The proof of Theorem 1.5 for $d\ge3$ (Sections 4–5, pp. 6–12) was read clause by clause and is incomplete as printed: the twelve source issues below were found, and a repaired chain that keeps the paper's method and constant is recorded on the repaired higher-dimensional chain page as a compilation-supplied, author-recorded reconstruction. One independent review of that reconstruction is filed under evidence/verify/gap_audit; it found the reconstruction sound at the stated constant for $d\ge3$ and the printed proof incomplete. No grader's acceptance is on file, so neither page supplies independently accepted proof coverage, and no result of this source carries a verification tier.

Contents

Section 1 (pp. 1–3) recalls Erdős's 1945 Conjecture 1.1 (probability at least $c/n$ that $n$ signed planar unit vectors sum into the closed unit disk), the even-$n$ counterexample and the corrected radius $\sqrt2$, Beck's Theorem 1.2 ($c_d n^{-d/2}$ at radius $\sqrt d$ for vectors of norm at most one in $\mathbb R^d$), the odd-$n$ conjecture of He, Juškevičius, Narayanan and Spiro, its disproof with an $O(n^{-3/2})$ bound in Hollom–Portier–Souza (2025), and that paper's Question 1.9, restated here as Question 1.4 (p. 2): for unit vectors $v_1,\ldots,v_n\in\mathbb R^d$ with $n\not\equiv d \pmod2$, are there always signs with $\lVert\sum_i\eta_iv_i\rVert\le \sqrt{d-1}$? Throughout, norms are Euclidean (p. 1), $\xi_i$ are the random signs and $\eta_i$ deterministic ones, and an $r$-approximating sequence is one whose zonotope $Z(V)$ is within squared distance $r$ of the signed-sum set $S(V)$ at every point (Definition 2.3, p. 3).

• Theorem 1.3 (p. 2, proof p. 5): with $c=1/20$, $v_n=(1,0)$ and $v_{2i-1}=v_{2i}=(\cos\theta_i,\sin\theta_i)$, $\theta_i=\arcsin c^i$, the odd-$n$ signed sum lies in the closed unit disk with probability exactly $2^{-\lfloor n/2\rfloor}$. This is the construction that Hollom–Portier–Souza report on their p. 3 as a personal communication.
• Theorem 1.5 (p. 2, proof pp. 6–12): for every $d$ there is $\varepsilon= \varepsilon(d)>0$, and one may take $\varepsilon=2^{-100}d^{-80}$, such that unit vectors in $\mathbb R^d$ with $n\not\equiv d\pmod2$ have signs with $\lVert\sum_i\eta_iv_i\rVert\le\sqrt{d-\varepsilon}$. Page 2 notes that for $n\equiv d\pmod2$ an odd number of copies of each basis vector gives probability zero at every radius below $\sqrt d$, so the parity condition matters in every dimension, and that the $\sqrt{d-1}$ bound of Question 1.4 is known in two dimensions.

Section 2 (pp. 3–5) collects the tools: Remark 2.2 ($\operatorname{Conv} (S(V))=Z(V)$), Lemma 2.4 (Beck: vectors of norm at most one are $d$-approximating), Lemma 2.5 (eliminating vectors from a non-approximating sequence), Lemma 2.6 (the convex hull of $S(V)$ is a union of translated parallelotope cells), Fact 2.7 and Lemma 2.8 (a $\delta$-almost orthogonal unit sequence is within $3\delta^{1/2}d$ of an orthonormal basis, by a polar decomposition). Section 4 (pp. 6–9), titled "Bounds for $d\ge3$", deduces Theorem 1.5 from Lemma 4.1 (a dichotomy: $d+1$ unit vectors are $(d-\varepsilon)$-approximating or $\zeta$-almost orthogonal, $\zeta=18\varepsilon^{1/4}d^4$) and Lemma 4.2 (stability: a correlated pair or a large coefficient improves Beck's $d$-approximation), splitting into an oblique-pair case and a clustered case. Section 5 (pp. 9–12) proves the two lemmas. Section 6 (p. 12) repeats Question 1.4, exhibits for $d=3$, $n=4$ a family of tight examples with minimum signed-sum norm $\sqrt2$, and suggests $n=d+1$ as a starting point.

Source issues

The issues below were found on the page images at filing and are identified by page. Each is recorded with its effect on the argument; none is attributed to an author erratum, and none affects the statement of Theorem 1.3 or of Theorem 1.5.

• HS-01 (p. 1). The even-$n$ counterexample with $n/2$ copies each of $(1,0)$ and $(0,1)$ has a signed sum equal to zero when $n/2$ is even, so it contradicts Conjecture 1.1 only when $n/2$ is odd. One copy of $e_1$ and $n-1$ copies of $e_2$ works for every even $n\ge2$. Introductory only.
• HS-02 (p. 4). The proof sketch of Lemma 2.5 sets $\eta_i=\lambda_i$ when $\lambda_i=\pm1$; with the plus convention of display (2.1) the canceling choice is $\eta_i=-\lambda_i$. The lemma stands.
• HS-03 (p. 4). Lemma 2.6's second display writes $p+\operatorname{Conv}(W)$ for $p+\operatorname{Conv}(S(W))$, and its proof places the moving point in a cell's vertex set rather than its convex hull. The cell decomposition stands, including for repeated vectors.
• HS-04 (p. 5). Lemma 2.8's displayed polar-decomposition estimate does not transparently track the Frobenius norm and its factor $\sqrt d$. A singular-value argument gives the column bound $d\delta$, which is what the repair uses.
• HS-05 (pp. 6, 10). Lemma 4.2's second bullet concludes that $(v_1,\ldots,v_d)$ is $(d-\delta)$-approximating when some $|\lambda_i|>\delta$; the premise depends on the target $\lambda$, so the conclusion holds for that fixed target only. An orthonormal basis with $\lambda=0$ has minimum squared error $d$, so the uniform reading is false. The fixed-target form is what the later argument needs.
• HS-06 (p. 8). The planar error line after display (4.3) prints $p_2$; the surrounding orthogonal decomposition requires $p_1$.
• HS-07 (p. 9). The cluster case bounds a short difference by $\sqrt2\zeta^{1/2}$; with clusters defined by inner product at least $1-\zeta^{1/4}$, Fact 2.7 gives $\sqrt2\zeta^{1/8}$, so the balanced short sum has squared norm at most $2d\zeta^{1/4}$, not $2d\zeta$. The stated constant absorbs the change.
• HS-08 (p. 10). The chord computation prints $r^2=1+(1-\lambda_1^2) \cos^2\theta$ and then $r^2=1-(1-\lambda_1)^2\cos^2\theta$; the identity is $r^2=1+(1-\lambda_1)^2\cos^2\theta\le2-\sin^2\theta$.
• HS-09 (pp. 11–12). Step 3 of the proof of Lemma 4.1 infers display (5.2), $\zeta/2\le y_i\le1-\zeta/2$, for every coordinate $i$ from one oblique pair; only one coordinate is supported (for example $y=(e_1+e_2)/\sqrt2$ has an oblique pair and zero later coordinates). One coordinate suffices for the cell and cube-center steps.
• HS-10 (p. 11). The transfer from the orthonormal basis $E$ back to $X$ moves the signed sum but not the target point, which lives in the other zonotope. Moving both costs at most $2d^2\sqrt\varepsilon$.
• HS-11 (pp. 6–7). Case 1 applies Lemma 2.5 with $W=V\setminus\{u,w\}$ and $k=d$, which needs $|W|\ge d+1$, that is $n\ge d+3$; at $n=d+2$ the sequence is used directly.
• HS-12 (pp. 6, 9). The proof begins at $n\ge d+1$ without the case $n\le d-1$ (parity excludes $n=d$), and the last sentence of Case 2 states uniform $(d-\varepsilon)$-approximation where the clustering argument constructs one signed sum of small norm, which is what Theorem 1.5 asks.

The rearrangement on p. 12 leading to display (5.3) was checked independently and is correct; an earlier provisional allegation of a missing factor four was withdrawn.

Relation to problem 395

Theorem 1.3 gives the exponential upper bound for the odd-$n$ unit-radius variant of the catalog question, matching the exponential lower bound $\tfrac14(0.525)^n$ of Hollom–Portier–Souza Theorem 1.6 up to the base. Theorem 1.5 concerns the higher-dimensional parity variant. Neither touches the exact radius-$\sqrt2$ question, which He–Juškevičius–Narayanan–Spiro Theorem 1.1 settles.

Bears on. #395 — the odd-$n$ unit-radius variant (Theorem 1.3) and the higher-dimensional parity variant (Theorem 1.5); not the exact catalog question.