Anon 2026 resolution erdos problem 38 sparse dyadic shift

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lemma_1: Finite multisets of shifts in [2^m], the union of whose supports has counting function O((log x)^6), whose averaged powers of the truncated right shift on C^N differ from the full average over [2^m] by at most 1/m in operator norm for every N at most 2^m.

theorem_1: A set B that is not an additive basis such that, for 0 < alpha < 1, every set A of Schnirelmann density alpha and every N >= 1, some b in B makes A together with A + b hold at least (alpha + f(alpha))N elements of {1, ..., N}, with f(alpha) > 0 given explicitly.

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“A resolution of Erdős Problem 38,” six-page manuscript, posted in the spicylemonade/erdos-38 repository as 38.pdf. The PDF metadata gives creation date 25 April 2026 and no author; the site discussion credits GPT 5.5 Pro for the solution and Liam Price for a later cleanup. The copy read for this card is that file, six pages, 223,509 bytes, read page by page. No notice is printed in the six-page manuscript, and the repository that posted it has no license file and no license in GitHub's record (https://github.com/spicylemonade/erdos-38, read 2026-10-02); the term is unstated.

The manuscript constructs one set $B$ with $|B\cap[1,x]|=O((\log x)^6)$, so $B$ is not an additive basis, while every set $A$ of Schnirelmann density $\alpha\in(0,1)$ and every $N\geq1$ have a shift $b\in B$ satisfying

$$|(A\cup(A+b))\cap[N]|\geq(\alpha+f(\alpha))N,$$

where, with $\beta=1-\alpha$,

$$m_0(\alpha)=\left\lceil\frac{16}{\alpha\beta^2}\right\rceil, \qquad f(\alpha)=\min\left\{\frac{\beta}{2}, \frac{\alpha\beta^2}{32},2^{-m_0(\alpha)}\right\}.$$

The shifts are fixed by a random choice of sparse multisets of dyadic shifts whose averages approximate the full dyadic shift average in $L^2$ operator norm; a counting argument over the shifts then finds the increment. The manuscript has two labeled results:

• [[integer_sequences/anon_2026_resolution_erdos_problem_38_sparse_dyadic_shift/lemma_1|Lemma 1: sparse dyadic shift averages]] (Section 1; statement pp. 1–2, proof pp. 2–3).
• [[integer_sequences/anon_2026_resolution_erdos_problem_38_sparse_dyadic_shift/theorem_1|Theorem 1: a sparse non-basis with uniform density increments]] (statement p. 1, proof in Section 2, pp. 4–6).

Read status. Claims checked: the statements of Lemma 1 and Theorem 1 and the definitions on p. 1 were read clause by clause on the print. The proofs were read for structure, and the result pages give sketches in the corpus's words; no proof is recorded here as verified.

Bears on. #38: Theorem 1 asserts a set $B$ of the kind the problem asks for, with an explicit $f(\alpha)>0$ for $0<\alpha<1$; Lemma 1 supplies the sparse shifts it is built from. The problem's acceptance record is kept on its problem page.

Source identity and limits. The manuscript names no author. The current FormalConjectures source for Problem 38 still ends its theorem with a Lean sorry. The pages above state the manuscript's results and sketch its proofs; they do not claim a checked Lean formalization.

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