Let be a function on some measure space with measure (for instance or a finite set).
Often one is interested in finding lower bounds for the norm of , that is , but has no way to directly estimate it. As a toy example, we can consider via . Estimating the norm directly seems hard.
But sometimes, we are able to estimate higher norms of . This is useful to our original problem, since an application of Holder’s inequality reveals that a lower bound on the norm and an upper bound on the norm gives a lower bound on the . To see this, note .
We can apply this idea to our original example. Parseval’s identity gives that while orthogonality and the divisor bound give that . This gives and is expected by the heuristic that a typical exponential sum should be about square root of the length of the sum.
My intuition is the following. Suppose the measure space is a probability space. Then . We are basically trying to reverse this inequality. Equality holds when is constant, that is is not too concentrated. The upper bound on the norm of implies that indeed is not too concentrated.
We mention that there is nothing too special about the exponents 2 and 4 chosen for the above discussion (although they are convenient for the specific example I chose).