By allowing interactions between electrons, a new Harper's equation is derived to examine the effects of electron correlations on the Hofstadter energy spectra. It is shown that the structure of the Hofstadter butterfly for the system of correlated electrons is modified only in the band gaps and the band widths, but not in the characteristics of self-similarity and the Cantor set.

A possibility of holon (boson) pair condensation is explored for hole doped high Tc cuprates, by using the U(1) slave-boson representation of the t-J Hamiltonian with the inclusion of hole-hole repulsion. A phase diagram of the hole doped high Tc cuprates is deduced by allowing both the holon pairing and spinon pairing. It is shown that the spin gap size remains nearly unchanged below the holon pair condensation temperature. We find that the s-wave holon pairing under the condition of d-wave singlet pairing is preferred, thus allowing d-wave hole pairing.

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