Nonaveraging sets methods and theorems

Split x+y

a(x+y)<=2a(x)+a(y)-1 if x>=y.

Product x*y

m(x*y)<=m(x)*m(y).

Wrap

m(x)<=2a(x)-1.

Drop

m(x)<=m(x+1).

Theorem N

Let k₁,k₂,k₃,...,k_N be a nonaveraging modular solution mod P, an increasing sequence of positive integers. Then for r=1,2,3,...,N
a(N*(q-1)+r)<=P*(a(q)-1)+k_r.
Note 1. If we have one modular solution, we can produce all its linear transformations, as there may be different particular solutions suitable for various r.
Note 2. Modular solutions are not turned automatically into theorems at the moment.
Note 3. It is unlikely but possible that a modular solution that is not improving known bound on m(n) (and therefore not saved by verification program) can still be useful.

Search

Direct search if everything above has failed. Sometimes this is like guesswork, as in case of m(216)<=3889, where I took 216-th roots of unity (or 18th nonzero powers) mod 3889.