Wiki
Wiki

Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated


Claim. For every constant c>0c>0 and every sufficiently large prime pp, every A⊆Fp∖{0}A\subseteq\mathbb F_p\setminus\{0\} with ∣A∣≤ec(log⁡p)1/4|A|\le e^{c(\log p)^{1/4}} has an ordering whose partial sums are distinct, indeed a two-sided valid one, in which no proper nonempty run of consecutive terms sums to zero: the question of Problem 475 for sets of that size. This is Theorem 1.2 (p. 1) of B. Bedert and N. Kravitz, Graham's rearrangement conjecture beyond the rectification barrier, which also holds, with a nearly identical proof, in every abelian group with no nonzero element of order smaller than pp. The method: a structure theorem splits the set into large dissociated sets and a rectifiable remainder; the remainder is ordered inductively as in Kravitz's integer argument, and the dissociated sets receive random orderings. The result widens Kravitz's range t≤log⁡p/log⁡log⁡pt\le\log p/\log\log p for large primes. Read depth: Theorem 1.2 and the proof sketch of Section 1.2 are checked in the arXiv version (v2, 7 January 2025, which incorporates the referee's suggestions); the proof (Sections 3--6) is not read.

Covers. For each c>0c>0: every prime beyond a threshold depending on cc, which the paper does not state, and every AA with ∣A∣≤ec(log⁡p)1/4|A|\le e^{c(\log p)^{1/4}}. No explicit prime is covered.

Depends on. Nothing in this wiki: the result is the paper's own, filed on its library result page. Kravitz's integer ordering theorem (Theorem 1.3 of arXiv:2407.01835), which the proof uses as a step, enters as a cited input of this refereed paper, not through the pending Kravitz page.

Acceptance. Refereed publication: Israel Journal of Mathematics 273 (2026), no. 1, 471--500, published online 30 November 2025 (the journal text is not held and not compared with the arXiv version). The site's commentary credits the range, but its label DECIDABLE leaves the problem open, so that credit is not reviewed evidence.