Let $A\subset \mathbb{N}$ be such that every large integer can be written as $n^2+a$ for some $a\in A$ and $n\ge 0$. What is the smallest possible value of$$\mathrm{limsup}\frac{|A\cap \{1,\dots ,N\}|}{N^{1/2}}?$$Is$$\mathrm{liminf}\frac{|A\cap \{1,\dots ,N\}|}{N^{1/2}}>1?$$

Erdős 33 (smallest limsup |A∩[1,N]|/√N for additive complements of squares) is open; exact value unknown. I confirmed the record sandwich inf ∈ [4/π, 2φ^{5/2}]≈[1.273, 6.660], gave a clean self-contai

Erdős 33 (smallest limsup |A∩[1,N]|/√N for additive complements of squares) is open; exact value unknown. I confirmed the record sandwich inf ∈ [4/π, 2φ^{5/2}]≈[1.273, 6.660], gave a clean self-contained proof that L(W) ≥ 4/π holds for the LIMSUP of every complement (Riemann-sum argument, π/4 constant machine-verified), and argued the 2025 "no exact-on-average" excess (≫√N(log N)^δ) is o(N) so cannot lift the lower constant above 4/π. Computationally: finite/horizon-greedy "complements" are invalid (explicit failure at k=3,003,287); valid greedy and naive constructions cannot beat van Doorn's 6.66. No new record; honest partial.