On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime

Given a finite family F of linear forms with integer coefficients, and a compact abelian group G, an F-free set in G is a measurable set which does not contain solutions to any equation L(x)=0 for L in F. We denote by d_F(G) the supremum of m(A) over F-free sets A in G, where m is the normalized Haar measure on G. Our main result is that, for any such collection F of forms in at least three variables, the sequence d_F(Z_p) converges to d_F(R/Z) as p tends to infinity over primes. This answers an analogue for Z_p of a question that Ruzsa raised about sets of integers.