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Density increments from shifts by nonbasis sets

Can one sparse set B, itself not an asymptotic additive basis, always supply a shift b that increases the density of A∪(A+b) by an amount depending only on the positive Schnirelmann density of A? The finite-cutoff target asks for at least (α+f(α))N covered points with f(α)>0.

Why it matters and what counts as progress

Why it matters: A single useful shift from a sparse non-basis would show that additive density improvement does not require basis structure.

What counts as progress: Construct such a non-basis B with a uniform positive density increment or sharpen the dependence on α and the cutoff.

How the problem developed

  1. 1935 · Partial result [1]

    Paul Erdős Proved a density-increment result when the shift set is a basis; the later problem asks whether a positive increment persists for nonbasis sets.

  2. 1956 · Open problem [2]

    Paul Erdős Raised the extension from basis sets to nonbasis sets in his additive number theory problems and results paper.

  3. 25 Apr 2026 · Reported claim [2][3]

    GPT-5.5 Pro, prompted by a user and forum participants A manuscript claims the requested positive density increment for every nonbasis shift set; Thomas Bloom accepted the solution claim, with checking discussion recorded on the problem page.

AI contribution

Sources

  1. Paul Erdős, On the arithmetical density of the sum of two sequences, one of which forms a basis for the integers
  2. Erdős Problem 38
  3. Erdős Problem 38 manuscript and Lean formalization
  4. Erdős #38 discussion
  5. Six-page Erdős #38 manuscript
  6. Community Lean proof of Erdős #38
  7. Erdős #38 source reconstruction
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