  Given a sequence of tokens, such as words, the task of next-token prediction
is to predict the next-token conditional probability distribution. Decoder-only
transformers have become effective models for this task, but their properties
are still not fully understood. In particular, the largest number of distinct
context sequences that a decoder-only transformer can interpolate next-token
distributions for has not been established. To fill this gap, we prove upper
and lower bounds on this number, which are equal up to a multiplicative
constant. We prove these bounds in the general setting where next-token
distributions can be arbitrary as well as the empirical setting where they are
calculated from a finite number of document sequences. Our lower bounds are for
one-layer transformers and our proofs highlight an important injectivity
property satisfied by self-attention. Furthermore, we provide numerical
evidence that the minimal number of parameters for memorization is sufficient
for being able to train the model to the entropy lower bound.
