Diophantine problems in variables restricted to the values 0 and 1

AbstractLet Fx1,…,xs be a form of degree d with integer coefficients. How large must s be to ensure that the congruence F(x1,…,xs) ≡ 0 (mod m) has a nontrivial solution in integers 0 or 1? More generally, if F has coefficients in a finite additive group G, how large must s be in order that the equation F(x1,…,xs) = 0 has a solution of this type? We deal with these questions as well as related problems in the group of integers modulo 1 and in the group of reals