The pigeonhole principle says that you cannot use pi to compress all data. You could certainly use this scheme to compress some data. For example, if you were

The pigeonhole principle says that you cannot use pi to compress all data. You could certainly use this scheme to compress some data. For example, if you were storing a file containing the contents of pi (from the beginning), it would compress very well with this scheme. This is the same tradeoff that all compression schemes make. gzip can compress some files, but others will grow. The hope is that the domain of files you actually want to apply it to will shrink instead of grow. That said, given that the digits of pi are more or less uniformly distributed, there is no particular reason to believe that common bit patterns will be likely to appear sooner in pi than uncommon bit patterns. So it is unlikely that it will compress very many of the files you'd have sitting around on your hard drive. But the pigeonhole principle alone does not prove this.