From $K(n+1)_*(X)$ to $K(n)_*(X)$

Let $X$ be a space of finite type. Set $q=2(p-1)$ as usual, and define the mod $q$ support of $K(n)^*(X)$ by $S(X,K(n)) = \{ m \in { \mathbb Z}/q{ \mathbb Z}\mid \bigoplus_{d \equiv m \operatorname{mod}q} K(n)^d \neq 0 \}$ for $n>0.$ Call $K(n)^*(X)$sparse if there is no $m \in { \mathbb Z}/q{ \mathbb Z}$ with $m, m+1 \in S(X,K(n)).$

Then we show the relation $S(X,K(n)) \subseteqq S(X,K(n+1))$ for any finite type space $X$ with $K(n+1)^*(X)$ being sparse.

As a special case, we have $K(n+1)^{odd}(X) = 0 \Longrightarrow K(n)^{odd}(X) = 0,$ and the main theorem of Ravenel, Wilson and Yagita is also generalized in terms of the mod $q$ support.