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We prove that there exists $A\subseteq \mathbb{N}$ such that \[ R_A(N)=\frac{\pi}{4}N+ O\!\left(N^{1/4}\sqrt{\log N}\right), \] where $R_A(N)=\#\{(a,b)\in A^2:a+b\le N\}$. This improves the record $O(N^{1/4}\log N)$ obtained by Ruzsa in 1997.
We prove that for additive complements $A$ and $B$, if $$ \limsup_{x\to+\infty}\frac{A(x)B(x)}{x}<\frac43, $$ then $$ A(x)B(x)-x\rightarrow+\infty\qquad(x\to+\infty). $$ This improves the earlier upper bound $3-\sqrt{3}$ due to Fang and Chen.
For integers $d\geq 3$, let $F_{2,d}(n)$ be the largest size of a subset of $[n]$ containing no two distinct elements whose product is a perfect $d$-th power, and let $f_{2,d}(n)$ denote the analogous quantity when the two elements need not be distinct. Fleiner, Juh\'asz, K\"ov\'er, Pach, and S\'andor proved that both...
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