2026/07/21 by Minkyu Jung
#math.CO #math.NT
For a finite set A⊂ℤ, write hA for its h-fold sumset, and let R(h,k)=\|hA|:A⊂ℤ, |A|=k\. We determine the part of R(h,4) lying between 4h+2 and 6h-4: for h=4 the only value is 5h-1, while for h≥ 5 the only values are 5h-1 and 5h+1. This proves Rajagopal's conjectured gap 5h∉ R(h,4) for every h≥ 4. For h≥ 6, it also yields the new missing interval [5h+2,6h-4], which lies outside Rajagopal's general excluded set. Lev's lower bound for the successive growth of multiple sumsets reduces the problem to normalized sets of affine diameter five, of which there are only six. Reflection and four elementary exact sumset computations finish the classification.