Timeline for Bounds on how many Sidon sets required to cover an integer range from 0-N

9 events

when toggle format	what		by	license	comment
Jan 9 at 19:42	vote	accept	Bobby Morelli
Jan 4 at 19:10	comment	added	Sandeep Silwal		beautiful answer!
Jan 4 at 17:22	history	edited	Emil Jeřábek	CC BY-SA 4.0	added 525 characters in body
Jan 4 at 17:20	comment	added	Emil Jeřábek		Your suggestion is on the right track, though. Whatever the maximal gap size is, I believe $p+1$ translates (non-consecutive) of Singer’s set do cover the whole cyclic group; I’ve updated the answer.
Jan 4 at 17:16	history	edited	Emil Jeřábek	CC BY-SA 4.0	added 525 characters in body
Jan 4 at 14:33	comment	added	Sean Eberhard		Good point, that argument doesn't quite work.
Jan 4 at 14:11	comment	added	Emil Jeřábek		Ah, this is good to know. Though I don’t quite follow the argument. Why cannot there be a gap of size larger than about $\sqrt N$? I can see that if there is a gap of size $g$, then we get a Sidon set of size $p+1$ in $[0,p^2+p+1-g)$, which implies $p+1<\sqrt{p^2+p+1-g}+\sqrt[4]{p^2+p+1-g}+1$ by the Lindström bound. But this only bounds $g$ by $2p^{3/2}\sim2N^{3/4}$ or so, not $N^{1/2}$.
Jan 4 at 12:12	comment	added	Sean Eberhard		One can improve this to $\sim \sqrt N$. Choose $p$ minimal so that $p^2+p+1 > N$ and use the Singer construction of a perfect difference in the cyclic group of order $p^2+p+1$. There cannot be a gap larger than about $N^{1/2}$, by the Erdős--Turan bound.
Jan 4 at 11:29	history	answered	Emil Jeřábek	CC BY-SA 4.0