I am looking for a sequence of topological spaces $X_n$, $n\in\mathbb N$, with the following property. Let $\tilde{K}^0(X_n)$ be the complex reduced $K$-theory group of $X_n$ (with respect to some choice of base point). I would like there to be a class $\xi_n\in\tilde{K}^0(X_n)$ such that $\xi$ cannot be represented as the difference of two vector bundles of rank at most $n$.

Question: How could one choose/construct $X_n$ and $\xi_n$ for all $n$?

Comments: I thought one might be able to take $X_n=S^n$ (say for $n$ even), which has a Bott vector bundle $\beta_n$ whose dimension increases with $n$. The class of $\beta_n-1$ generates $\tilde{K}^0(S^n)$, and I suspect it cannot be represented by the difference of two lower-dimensional bundles, but I'm not sure if this is true or how to show it.

I am looking for a sequence of topological spaces $X_n$, $n\in\mathbb N$, with the following property. Let $\tilde{K}^0(X_n)$ be the complex reduced $K$-theory group of $X_n$ (with respect to some choice of base point). I would like, for each $n$, for there to be a class $\xi_n\in\tilde{K}^0(X_n)$ such that $\xi_n$ cannot be represented as the difference of two vector bundles of rank at most $n$.

Question: How could one choose/construct $X_n$ and $\xi_n$ for all $n$?

Comments: I thought one might be able to take $X_n=S^n$ (say for $n$ even), which has a Bott vector bundle $\beta_n$ whose dimension increases with $n$. The class of $\beta_n-1$ generates $\tilde{K}^0(S^n)$, and I suspect it cannot be represented by the difference of two lower-rank bundles, but I'm not sure if this is true or how to show it.