The structure of higher sumsets

Merging together a result of Nathanson from the early 70s and recent Granville Walker, we show that for any finite set A A integers with alttext="min left-parenthesis upper A right-parenthesis equals 0"> min stretchy="false">( stretchy="false">) = 0 encoding="application/x-tex">\min (A)=0 alttext="gcd 1"> form="prefix">gcd 1 encoding="application/x-tex">\gcd (A)=1 there exist two sets, “head” “tail”, such if alttext="m greater-than-or-equal-to max minus StartAbsoluteValue EndAbsoluteValue plus 2"> m ≥ form="prefix">max −<!-- − <mml:mrow class="MJX-TeXAtom-ORD"> stretchy="false">| + 2 encoding="application/x-tex">m\ge \max (A)-|A|+2 , then alttext="m"> encoding="application/x-tex">m -fold sumset encoding="application/x-tex">mA consists union head, appropriately shifted tail, long block consecutive separating them. We give sharp estimates length block, find all those sets which bound alttext="max encoding="application/x-tex">\max cannot be substantially improved.