Explanation of a variable in this ~1page paper on arithmetical progressions

Question

I'm a first year so this is purely out of curiosity. This paper https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1078964/?page=1 seems to propose a better density for arithmetical-progression free sets. I think I understand the construction except for the variable k. Why does k need to be $\le n(d-1)^2$? From the requirement norm(A)=k, it seems that I can choose k from 0 to $\lfloor d\sqrt n \rfloor$ to be more restrictive, allowing for a better bound when later the pigeonhole principle is used to establish that for some k $S_k(n,d)$ must contain at least a certain number of terms. (That's what's happening right?)