A lower bound for additive complements to the primes

@misc{radomskii2026lower,
  title = {{A lower bound for additive complements to the primes}},
  author = {Artyom Radomskii},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14127},
  eprint = {2610.14127},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14127},
}

TY  - UNPB
TI  - A lower bound for additive complements to the primes
AU  - Artyom Radomskii
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.14127
UR  - https://arxiv.org/abs/2610.14127
AB  - We prove that every set ℬ⊆ℤ_≥0 for which ℙ+ℬ contains all sufficiently large integers satisfies B(X)≫log Xloglog X, where B(X)=#(ℬ∩[0,X]) and the implied constant is absolute. This proves a quantitative form of Ruzsa's Conjecture 2 and gives a negative answer to the O(log X) question in Erdős Problem 32. The bound follows from a more general result: for an arbitrary set ℬ⊆ℤ_≥0 and sufficiently large X, we construct a block of ⌊ clog Xloglog X/(B(X)+1)⌋ consecutive integers in [X/2,X] outside ℙ+ℬ, provided this length is positive. As an application, we obtain G_ℛ(X)≫loglog X for the longest block in [1,X] containing no Romanoff number p+2^n, improving the bound G_ℛ(X)≫loglog X/logloglog X of Kalmynin and Konyagin. The argument also applies to p+a^n for every fixed integer a≥2. The key step is a lemma on large shifts, proved using truncated products of divisor sums and the removal of auxiliary primes dividing differences of shifts.
ER  -