Regularity lemma proof by Croot, function question (Theorem 2 proof, p. 4) The proof of Theorem 2, on page 4 of Croot's proof of the Regularity Lemma has the following function definition: $$ f(t) = t \sum_{1 \le j \le t}{|P_{((j-1)k/t,jk/t]}|^2} $$ I understand that this is summing the squares of the number of items in each interval, **but I don't understand where the $t$ outside the summation comes from**. From my understanding, without the $t$, the summation is already adding all $t$ intervals. The only reasonable thing I've come up with, is that it accounts for the intervals in-between the original intervals (e.g., it starts at 0, and then everything shifts to the right $t$ times to cover the entire space). Is that right? If not, where did this outer $t$ come from?

The $f(t)$ here is supposed to act like a sort of norm by which to judge partitions. A finer partition should have a larger norm, and using $f(t)$ as the norm, we get this property. Without that extra $t$ out front, this property no longer holds. The reason the proofs of regularity-type lemmas work is that the properly defined norm strictly increases by a constant amount after each refining of the partition whenever the substructures are not $\epsilon$-regular, and that the norm is bounded above. Either the substructures are $\epsilon$-regular and we are done, or we may refine the partition and increase the norm. But we cannot refine the partition forever since the norm is bounded above, so at some stage the substructures are $\epsilon$-regular. Thus $f(t)$ is chosen carefully to have those two properties.

On a more intuitive level, you can think of the extra $t$ out front as some "normalization" factor.