Linear extension of the Erdős–Heilbronn conjecture

The famous Erdős–Heilbronn conjecture plays an important role in the development of additive combinatorial number theory. In 2007 Z.W. Sun made the following further conjecture (which is the linear extension of the Erdős–Heilbronn conjecture): For any finite subset A of a field F   and nonzero elements a1,…,ana1,…,an of F, we have|{a1x1+⋯+anxn:x1,…,xn∈A,andxi≠xjifi≠j}|⩾min{p(F)−δ,n(|A|−n)+1}, where the additive order p(F)p(F) of the multiplicative identity of F   is different from n+1n+1, and δ∈{0,1}δ∈{0,1} takes the value 1 if and only if n=2n=2 and a1+a2=0a1+a2=0. In this paper we prove this conjecture of Sun when p(F)⩾n(3n−5)/2p(F)⩾n(3n−5)/2. We also obtain a sharp lower bound for the cardinality of the restricted sumset{x1+⋯+xn:x1∈A1,…,xn∈An,andP(x1,…,xn)≠0}, where A1,…,AnA1,…,An are finite subsets of a field F   and P(x1,…,xn)P(x1,…,xn) is a general polynomial over F.