Hollom 2025 double jump phase transition reverse littlewood offord

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theorem_1_13: Three equal blocks of the vertices of an inscribed equilateral triangle have a signed sum of norm at most root two with probability (1+o(1)) times 2 root 3 over pi n, answering questions of Beck and of He, Juškevičius, Narayanan and Spiro negatively; recorded at statement depth.

theorem_1_14: In every sufficiently large dimension d, for large n some orthogonal-type set has a smaller probability of a signed sum of norm at most root d than every simplicial-type set, by the factor 2^{-0.005 d}; recorded at statement depth.

theorem_1_15: There is an absolute delta below one such that in every dimension d at least 2, for large n, some set of n unit vectors beats every orthogonal-type set at radius root d by the factor delta; recorded at statement depth.

theorem_1_4: For every delta above zero and odd n, planar unit vectors have a signed sum of norm at most 1 plus delta with probability at least c_delta over n; recorded at statement depth.

theorem_1_6: For odd n, planar unit vectors have a signed sum in the closed unit disk with probability at least one quarter times 0.525 to the n; recorded at statement depth.

theorem_1_7: For every odd n some planar unit vectors have a signed sum in the closed unit disk with probability at most C over n to the three halves, so the odd-n unit-radius conjecture fails; recorded at statement depth.

theorem_1_8: For d at least 2 and every n of parity opposite to d, some n unit vectors in d-space have a signed sum of norm at most root of d minus one with probability O(n^{-(d+1)/2}); the printed domain d at least 1 fails at d equals 1.

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Lawrence Hollom, Julien Portier, and Victor Souza, Double-jump phase transition for the reverse Littlewood–Offord problem. arXiv:2503.24202v1 [math.CO], 31 March 2025, 32 pages.

Source identity

The copy read for this card is the arXiv v1 manuscript (watermark "arXiv:2503.24202v1 [math.CO] 31 Mar 2025" on p. 1), 32 physical pages whose printed numbers equal the PDF page numbers (435,852 bytes). Provenance: downloaded from https://arxiv.org/abs/2503.24202 on 2026-09-05. A publication record in the survey download set of September 2026 lists the paper in the Journal of the London Mathematical Society (2026), DOI 10.1112/jlms.70539; that version was not acquired and no version of record was compared, so every locator below is to arXiv v1. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2503.24202), every other right reserved.

Read status: claims checked for Theorems 1.4, 1.6, 1.7, 1.8, 1.13, 1.14 and 1.15 on the page images; all 32 pages were read at filing. The proofs were read but not reconstructed and not independently reviewed; no proof coverage is claimed for any result, and the statement notes below are filing observations, not review verdicts.

Contents

Section 1 (pp. 1–6) recalls Erdős's 1945 conjecture (Conjecture 1.1) that signed sums of $n$ unit complex numbers lie in the closed unit disk with probability at least $c/n$, its failure for even $n$ (an odd number of copies each of $(1,0)$ and $(0,1)$, p. 1), Beck's 1983 theorem at radius $\sqrt d$ in $\mathbb R^d$ (Theorem 1.2, p. 2), and the conjecture of He, Juškevičius, Narayanan and Spiro that the unit-radius statement holds for odd $n$ (Conjecture 1.3, p. 2, their Conjecture 4.1). The paper's results for odd $n$ in the plane, and Theorem 1.8 in $\mathbb R^d$, all for independent uniform signs $\varepsilon_i$ and $\sigma_V=\sum_i\varepsilon_iv_i$:

• Theorem 1.4 (p. 2, proof pp. 8–10): for every $\delta>0$ and odd $n$, $\Pr(\lVert\sigma_V\rVert_2\le1+\delta)\ge c_\delta/n$.
• Theorem 1.6 (p. 2, proof pp. 10–15): for odd $n$, $\Pr(\lVert\sigma_V\rVert_2\le1)\ge\tfrac14(0.525)^n$.
• Theorem 1.7 (p. 3, proof pp. 15–16 as the $d=2$ case of Theorem 1.8): for every odd $n$ some planar unit vectors have $\Pr(\lVert\sigma_V\rVert_2\le1)\le Cn^{-3/2}$, which disproves Conjecture 1.3.
• Theorem 1.8 (p. 4, restated and proved pp. 15–16): the $d$-dimensional construction at radius $\sqrt{d-1}$ with probability $O_d(n^{-(d+1)/2})$ for $n\not\equiv d\pmod 2$; its printed domain $d\ge1$ must read $d\ge2$, see the result page.

The paper reports on p. 3 that Gregory Sorkin, after a seminar on this work, found planar unit vectors for odd $n$ with $\Pr(\lVert\sigma_V\rVert_2\le1)\le2^{-(n-1)/2}$; the reference is a personal communication (reference [23], p. 27), and no proof is printed. Its published form is Hollom–Sorkin (2025), Theorem 1.3. Page 3 collects the resulting picture for $F_{2,r}(n)=\inf_V\Pr(\lVert\sigma_V\rVert_2\le r)$ with $n$ odd: zero for $r<1$, between $\Omega(0.525^n)$ and $O((1/\sqrt2)^n)$ at $r=1$, and $\Theta_r(n^{-1})$ for $r>1$, the "double jump" of the title.

Section 6 (pp. 17–24) concerns which unit vectors minimize $\Pr(\lVert\sigma_V\rVert_2\le\sqrt d)$:

• Theorem 1.13 (p. 5, proof pp. 17–18): three equal blocks of the vertices of an inscribed equilateral triangle have probability $(1+o(1))\,2\sqrt3/(\pi n)$ at radius $\sqrt2$, below the $(1+o(1))\,4/(\pi n)$ of the orthogonal pair (p. 5). Page 5 states that this gives a negative answer to Beck's Question 1.10, to the second part of Question 1.11 (He–Juškevičius–Narayanan–Spiro Question 4.2), and disproves Conjecture 1.12 (their Conjecture 4.3).
• Theorem 1.14 (p. 6, proof p. 23): in every sufficiently large dimension, one orthogonal-type set beats every simplicial-type set by the factor $2^{-0.005d}$ for large $n$.
• Theorem 1.15 (p. 6, proof pp. 23–24): an absolute $\delta<1$ such that in every $d\ge2$, for large $n$, some set beats every orthogonal-type set by the factor $\delta$; for $d\ge3$ the set is of mixed type.

Section 2 (pp. 6–7) states the tools: Proposition 2.1 is the pairing estimate of He–Juškevičius–Narayanan–Spiro (their Proposition 2.1, quoted for $2n$ unit vectors), Proposition 2.2 is Robbins's form of Stirling's formula, Proposition 2.3 the asymptotic of $\sum_k\binom mk^q$ (attributed to Pólya–Szegő and Farmer–Leth), and Propositions 2.4–2.6 are binomial-convolution estimates proved in Appendix A (pp. 27–31). Appendix B (pp. 31–32) computes the exact minimal lattice counts $f_0(d)$ and $f_1(d)$ behind Proposition 6.4 (Proposition B.1), with a numerical check reported for $8\le t\le168$. Section 7 (pp. 24–26) poses Questions 7.1, 7.2, 7.4, 7.5, Problem 7.3 and repeats Question 1.9 (refined vector balancing: signs with $\lVert\sum\eta_iv_i\rVert_2\le \sqrt{d-1}$ when $n\not\equiv d\pmod2$), which is Hollom–Sorkin's Question 1.4.

Statement notes recorded at filing

These were noted while reading the page images. They are recorded so a later reconstruction does not rediscover them; none has been reviewed independently and none is attributed to an author erratum.

• Theorem 1.8 (pp. 4, 15) is printed for $d\ge1$. For $d=1$ every unit vector is $\pm1$, the radius is $\sqrt{d-1}=0$, and for even $n$ the event is $\sigma_V=0$, of probability $\binom n{n/2}2^{-n}$, which is not $O(n^{-1})$. The proof on pp. 15–16 uses the perturbed vectors $e_1^\pm=(\cos\beta,\pm\sin\beta,0,\ldots,0)$ and a second block on $e_2$, so it needs $d\ge2$. Theorem 1.7 ($d=2$) is unaffected.
• Proposition 2.4 (pp. 7, 27) prints the relative error as $O_\varepsilon(X/\ell^{1/2-\varepsilon})$. At $X=0$ this asserts an exact identity; for $q=2$, $m_1=m_2=m$ even and $x_1=x_2=0$ the left side is $\binom{2m}m$ by Vandermonde and the right side is $4^m/\sqrt{\pi m}$, which are never equal. An error term $O_\varepsilon((1+X)/\ell^{1/2-\varepsilon})$ is what the proof on pp. 28–29 supports.
• Proposition 2.6 (pp. 7, 31) never quantifies the shifts $x_i$; the proof uses $\binom{m_i}{(m_i+x_i)/2}\ge1$, which needs $m_i\ge|x_i|$ and $m_i\equiv x_i\pmod 2$.
• Corollary 6.5 (p. 19) claims one absolute constant for all $d\ge1$ and $n\ge1$, but its proof (p. 20) applies the $(1+o(1))$ asymptotic of Proposition 6.3(i), which is for fixed $d$ and $n\to\infty$. Theorem 1.14 uses it only for fixed $d$ and large $n$.
• Remark 6.2 (p. 18) lists the vectors $(1,0)$, $(-\sqrt3/2,1/2)$ and $(-\sqrt3/2,-1/2)$; these are not the vertices of an equilateral triangle. The proof of Theorem 1.13 on p. 17 uses $(1,0)$ and $(-1/2,\pm\sqrt3/2)$.
• The definitions of $r_c^*(d)$ in (1.2) on p. 4 and of $r_c(d)$ below it use $\liminf F_{d,r}(n)>0$ without the factor $n^{d/2}$ that Problem 7.3 on p. 25 includes; collinear unit vectors make $F_{d,r}(n)\to0$ for every fixed $r$, so the normalized reading is the one consistent with the stated values.
• In the proof of Theorem 1.4 (p. 10) the final display bounds $c_\delta/|A|$ below by $c_\delta/(n-40/\delta^2)$ using the lower bound on $|A|$; the bound $|A|\le n$ gives $c_\delta/|A|\ge c_\delta/n$ directly, which is all the theorem needs.

The base $0.525$ of Theorem 1.6 rests on $2^{-13/14}>21/40$ (p. 15, where $2^{-13/14}\approx0.5253$), which is equivalent to $40^{14}>21^{14}2^{13}$ and holds.

Relation to problem 395

Problem 395 asks the radius-$\sqrt2$ question for arbitrary $n$, which Theorem 1.2 (Beck) and the elementary proof of He–Juškevičius–Narayanan–Spiro Theorem 1.1 settle affirmatively. This paper does not change that status. It settles the odd-$n$ unit-radius variant negatively (Theorem 1.7), proves the approximate form at radius $1+\delta$ (Theorem 1.4), and by Theorem 1.13 disproves He–Juškevičius–Narayanan–Spiro Conjecture 4.3 and answers the second part of their Question 4.2 negatively; it leaves open which planar configurations minimize the radius-$\sqrt2$ probability (Question 7.2, p. 25).

Bears on. #395 — variants of the catalog question, not whether its radius-$\sqrt2$ probability is $\gg1/n$: for odd $n$, lower bounds $c_\delta/n$ at radius $1+\delta$ (Theorem 1.4) and $\tfrac14(0.525)^n$ at radius $1$ (Theorem 1.6), and a construction with probability $O(n^{-3/2})$ at radius $1$ that disproves He–Juškevičius–Narayanan–Spiro Conjecture 4.1 (Theorem 1.7); at radius $\sqrt2$, an upper bound $(1+o(1))\,2\sqrt3/(\pi n)$ on the least probability for $n=3k$, which by p. 5 disproves their Conjecture 4.3 (Theorem 1.13); and higher-dimensional analogues at radius $\sqrt{d-1}$ (Theorem 1.8) and $\sqrt d$ (Theorems 1.14 and 1.15).

No file of this source is held. The edition read, arXiv v1, carries only arXiv's non-exclusive distribution license; the version of record, Journal of the London Mathematical Society 113 (2026), no. 5, e70539, is under CC BY 4.0 according to its Crossref record (https://api.crossref.org/works/10.1112/jlms.70539), but it was not acquired, and the card cites the arXiv edition named above.