Exact diameter compression for four-point sumset cardinalities

Abstract

The least interval diameter needed to realize every attainable cardinality of an $h$-fold sumset of a four-element integer set is $\left(\genfrac{}{}{0ex}{}{h+2}{2}\right)+(h\mathrm{mod}2)$ for $2\le h\le 10$ and $d_h=(h-3)(h-1)$ for $h\ge 11$. We prove the formula uniformly for $h\ge 17$ and use complete finite searches, with a relation-rank coverage argument, for the remaining orders. For the lower bound, we show that every integer four-set with $\left(\genfrac{}{}{0ex}{}{h+3}{3}\right)-17$ distinct $h$-fold sums has normalized diameter in $\{d_h,d_h+1,d_h+2\}$ when $h\ge 9$, and we construct sets attaining $d_h$. For the upper bound, a relation-lattice index argument restricts larger minimum-diameter representatives to two or three active directions. Two general constructions and four special defects then provide short sets realizing every remaining cardinality.

arXiv comment19 pages, 4 tables. Code and exact computational data are provided as ancillary files