Bh[g]-Sequences with Large Upper Density Original Research Article

For any positive integersh, g, letm=m(h, g) be the largest possible integer such that for everyaset membership, variantZmthe equationa≡x1+x2+…+xh (mod m) has at mostgsolutions (up to rearrangement of thexiʹs). It is proved in this paper that there exists aBh[g]-sequenceBsubset of or equal to[1, n] with[formula]In particular, there exists aB2[2]-sequenceBsubset of or equal to[1, n] such that[formula]which improves an earlier result of Kolountzakis. It is also proved that for everyggreater-or-equal, slanted2 there exists an infiniteB2[g]-sequenceB={n1