Double-jump phase transition for the reverse Littlewood--Offord problem

TL;DR

This work advances the reverse Littlewood–Offord problem in the plane by showing that Erdős’ conjecture fails for odd $n$ via explicit constructions yielding $P( orm{oldsymbol{\sigma}}_2\,m{ obreakf ext{/≤}1)=O(n^{-3/2})$, while an approximate version at radius $1+ ext{δ}$ remains of order $1/n$. It also proves a tight exponential lower bound $P( orm{oldsymbol{\sigma}}_2\\le 1)\ge c(δ)^{n}$ and demonstrates a double-jump phenomenon at $r=1$, alongside refined results on optimal vector configurations (orthogonal, simplicial, mixed) that shape the lower bounds. The paper extends to higher dimensions, providing dimension-specific constructions and showing that simplicial and mixed configurations can outperform orthogonal ones, with parity playing a crucial role in the asymptotics. These results shed light on the intricate geometry of random signed sums, offer new thresholds and constructions, and open several directions for higher-dimensional and quantitative discrepancy questions with potential implications for related probabilistic and geometric problems.

Abstract

Erdős conjectured in 1945 that for any unit vectors $v_1, \dotsc, v_n$ in $\mathbb{R}^2$ and signs $\varepsilon_1, \dotsc, \varepsilon_n$ taken independently and uniformly in $\{-1,1\}$, the random Rademacher sum $σ= \varepsilon_1 v_1 + \dotsb + \varepsilon_n v_n$ satisfies $\|σ\|_2 \leq 1$ with probability $Ω(1/n)$. While this conjecture is false for even $n$, Beck has proved that $\|σ\|_2 \leq \sqrt{2}$ always holds with probability $Ω(1/n)$. Recently, He, Juškevičius, Narayanan, and Spiro conjectured that the Erdős' conjecture holds when $n$ is odd. We disprove this conjecture by exhibiting vectors $v_1, \dotsc, v_n$ for which $\|σ\|_2 \leq 1$ occurs with probability $O(1/n^{3/2})$. On the other hand, an approximated version of their conjecture holds: we show that we always have $\|σ\|_2 \leq 1 + δ$ with probability $Ω_δ(1/n)$, for all $δ> 0$. This shows that when $n$ is odd, the minimum probability that $\|σ\|_2 \leq r$ exhibits a double-jump phase transition at $r = 1$, as we can also show that $\|σ\|_2 \leq 1$ occurs with probability at least $Ω((1/2+μ)^n)$ for some $μ> 0$. Additionally, and using a different construction, we give a negative answer to a question of Beck and two other questions of He, Juškevičius, Narayanan, and Spiro, concerning the optimal constructions minimising the probability that $\|σ\|_2 \leq \sqrt{2}$. We also make some progress on the higher dimensional versions of these questions.

Figures (3)

• Figure 1: On the left, polygon $P$ with each term of the sum $u = v_1 - v_2 + v_3 - \dotsb + v_n$ highlighted. The linear span $U$ of the vector $u$ intersects the boundary of $P$ on $b$ and $-b$. In the middle, the (oblique) projections $\pi_U(\pm a_i)$ into $U$ are shown to pack in the segment $[-b,b]$. On the right, rotate so $U$ aligns with $x$-axis, reflect and relabel vectors so they are on the right half of the plane.
• Figure 2: The pairings involved in the characterisations of the case $\beta = -1$ (left) and the case $\beta = 1$ (right).
• Figure 3: The $\normstar{}$-ball of radius $\sqrt{1 - \abs{\beta}}$ centred at $u = (\beta,0)$ is fully contained in the unit $\norm{}_2$-ball centred at the origin.

Theorems & Definitions (71)

• Conjecture 1.1: Erdős
• Theorem 1.2: Beck
• Conjecture 1.3: He, Juškevičius, Narayanan, and Spiro
• Theorem 1.4
• Theorem 1.5: Swanepoel
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Conjecture 1.11: Conjecture 4.3 in He2024-cp
• Theorem 1.12