The large sieve inequality and additive decompositions of sums of squares

@misc{elsholtz2026large,
  title = {{The large sieve inequality and additive decompositions of sums of squares}},
  author = {Christian Elsholtz and Julia Stadlmann},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20554},
  eprint = {2610.20554},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20554},
}

TY  - UNPB
TI  - The large sieve inequality and additive decompositions of sums of squares
AU  - Christian Elsholtz
AU  - Julia Stadlmann
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20554
UR  - https://arxiv.org/abs/2610.20554
AB  - Building on the large sieve, we prove a new sieve estimate for sets which are missing various residue classes modulo prime squares. For certain interesting residue class configurations, our bound is a significant improvement over the corresponding Johnsen-Selberg sieve inequality. Our application of this sieve estimate is motivated by Ostmann's problem. We study additive decompositions of sums of squares and show that any non-trivial sumset decomposition must consist of two sets of roughly equal size: If A_1, A_2 ⊆ ℤ with A_1 + A_2 = {x^2 + y^2 : x, y ∈ ℤ} and |A_1|, |A_2| ≥ 2, then √x/(log x)^9/2 ≪ |A_i ∩ [1, x]| ≪ √x(log x)^4 for i ∈ {1, 2}.
ER  -