Diophantine problems in variables restricted to the values 0 and 1

Let $\text{F}$x₁,…,x_s be a form of degree d with integer coefficients. How large must s be to ensure that the congruence $\text{F}$(x₁,…,x_s) ≡ 0 (mod m) has a nontrivial solution in integers 0 or 1? More generally, if $\text{F}$ has coefficients in a finite additive group G, how large must s be in order that the equation $\text{F}$(x₁,…,x_s) = 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.