Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Graham's rearrangement conjecture

Huy Tuan Pham, Lisa Sauermann

Abstract

Graham conjectured in 1971 that for any prime $p$, any subset $S\subseteq \mathbb{Z}_p\setminus \{0\}$ admits an ordering $s_1,s_2,\dots,s_{|S|}$ where all partial sums $s_1, s_1+s_2,\dots,s_1+s_2+\dots+s_{|S|}$ are distinct. We prove this conjecture for all subsets $S\subseteq \mathbb{Z}_p\setminus \{0\}$ with $|S|\le p^{1-α}$ and $|S|$ sufficiently large with respect to $α$, for any $α\in (0,1)$. Combined with earlier results, this gives a complete resolution of Graham's rearrangement conjecture for all sufficiently large primes $p$.

On Graham's rearrangement conjecture

Abstract

Graham conjectured in 1971 that for any prime , any subset admits an ordering where all partial sums are distinct. We prove this conjecture for all subsets with and sufficiently large with respect to , for any . Combined with earlier results, this gives a complete resolution of Graham's rearrangement conjecture for all sufficiently large primes .
Paper Structure (7 sections, 19 theorems, 126 equations)

This paper contains 7 sections, 19 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Notation.
  3. Organization.
  4. Preliminaries
  5. Anticoncentration on boolean slices via Fourier Analysis
  6. Combinatorial anticoncentration deductions
  7. Rearrangement conjecture

Key Result

Theorem 1.2

For any $0<\alpha<1$, there exists a constant $C_\alpha>0$ such that the following holds. Let $p$ be a prime and let $S\subseteq \mathbb{Z}_p \setminus \{0\}$ be a subset with $C_\alpha\le |S|\le p^{1-\alpha}$. Then there exists a valid ordering of $S$, i.e. there is an ordering $s_1,s_2,\dots,s_{|S

Theorems & Definitions (43)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • proof
  • proof
  • proof
  • proof
  • proof
  • Lemma 3.1
  • ...and 33 more