## Thursday, May 03, 2007

Most problems were done pretty well; interestingly, only one person attempted #7, even though it has the shortest solution after #3. A number of people did #2 but no one really got part (b) exactly correctly. The issue there is that you have a real-valued (not integer-valued) random variable and you know that it has exponentially small tails; you want to show that the contribution to its $L_1$-norm from large numbers is exponentially small. There is a battle here, between, the contribution of $|u|$ and the probability $O(e^{-u^2/2})$, which of course the exponentially small quantity is going to win; still, one has to integrate, as shown in the solutions.
Also, nobody got #6(c) quite correct. The Tribes function is not exactly balanced; its empty Fourier coefficient is just known to be $\pm O(\log n / n)$. Thus $\mathbb{N}\mathbb{S}_{1/2 - \gamma}(T)$ will always be at most $1/2 - \Omega(\log^2 n / n^2)$, which is why the condition $\gamma \geq 1/n$ is necessary.