Kravitz 2024 rearranging small sets distinct partial sums

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theorem_1_2: Graham's rearrangement conjecture for sets of size at most log p over log log p, for every prime p, by rectification to the integers and an inductive ordering there.

theorem_1_3: The integer version of Graham's rearrangement conjecture, proved by induction on the size with the positive elements listed before the negative ones; with rectification it gives Theorem 1.2.

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Noah Kravitz, Rearranging small sets for distinct partial sums. arXiv:2407.01835 (2024).

Kravitz proves Graham's conjecture, repeated by Erdős, that every subset A of F_p \ {0} admits an ordering whose partial sums are all distinct, for all sets of size at most log p / log log p (Theorem 1.2); the paper's stated novelty is that this bound grows with p (p. 1), the results it cites covering |A| <= 12 and nonzero-sum sets of size p-2 or p-3. The proof has two short steps: Theorem 1.3 establishes the integer version, that every finite A ⊂ Z \ {0} has a valid ordering, by induction on |A| that lists the positive elements before the negative ones and, taking the side P of positive elements to have the larger or equal sum, removes an element of P whose removal makes the two sums unequal; then Lev's rectification theorem, which the paper quotes as Theorem 2.1 and calls an optimal refinement of Bilu, Lev and Ruzsa, Freiman-isomorphically transfers A ∪ {0} to a set of integers, and validity is expressible through non-equalities of sums of length at most |A| - 1, so the ordering pulls back. The method is rectification plus induction rather than the polynomial method and Combinatorial Nullstellensatz used for small cases. The paper bears directly on problem #475 (the Graham/Erdős conjecture on orderings of subsets of F_p with distinct partial sums), what the paper calls modest partial progress; the author notes Will Sawin obtained a similar result via the same two steps in a 2015 MathOverflow post, with different details.

The copy read for this card is arXiv:2407.01835v2 (18 August 2024, 4 pp.), whose pagination is used here; no journal version was located (Crossref bibliographic query, 2026-09-18). Read status: claims checked for Theorem 1.2, Theorem 1.3 and Lev's Theorem 2.1 (pp. 1--2, text layer), read clause by clause on 2026-09-18; the two half-page proofs (p. 2) were read in full and are not independently reviewed. Result pages: theorem_1_2 and theorem_1_3. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2407.01835), every other right reserved.

Source: https://arxiv.org/abs/2407.01835.

Bears on. #475: Theorem 1.2 (p. 1) answers the problem's question yes for every prime p and every A ⊆ F_p \ {0} with |A| <= log p / log log p, a partial result silent on larger sets, in a preprint with no journal version located. Theorem 1.3 (p. 1) is the integer version, which settles no case of the problem by itself and enters through Theorem 1.2.

Results. Page numbers are those of arXiv:2407.01835v2 (pp. 1--4).

• Theorem 1.2 (p. 1; proof p. 2): for a prime p, every A ⊆ F_p \ {0} with |A| <= log p / log log p has a valid ordering. Remark (3) (p. 3) extends it to subsets of size at most log p / log log p of G \ {0}, for an abelian group G with no nonzero element of order strictly smaller than p.
• Theorem 1.3 (p. 1; proof p. 2): every finite A ⊆ Z \ {0} has a valid ordering; the proof gives one with all positive elements before all negative ones.
• Theorem 2.1 (p. 2), quoted from Lev [14]: for ℓ ∈ N, a prime p and A ⊆ F_p with |A| <= ⌈log p / log ℓ⌉, A is ℓ-Freiman-isomorphic to a set of integers. It is recorded on the Theorem 1.2 page, not on a page of its own, since it is Lev's result.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.