Research proof index

adjointCarleson is the adjoint of carlesonOn.

For positive parameters $A,B,C,R$, the classical error-bound function $A\left(\frac{\log x}{R}\right)^B\exp\!\left(-C\sqrt{\frac{\log x}{R}}\right)$ is nonincreasing once $x\ge \exp\!\left(R(2B/C)^2\right)$.

Suppose that, for every error threshold $\delta>0$ and every measure tolerance $\varepsilon>0$, one can choose $N_0$ so that the set where $\sup_{N>N_0}\lVert f(x)-F_N(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$. Then $F_N(x)$ converges to $f(x)$ for almost every $x$.

This lemma proves the final algebraic step in the DG25 Theorem 3.1 proof. It shows that if R > e + 1, then e * (R / (R - 1)) < e + 1. The intuition is that the fraction R / (R - 1) is always greater than 1, but as R gets larger, it gets closer to 1. The hypothesis R > e + 1 provides a strong enough bound to ensure the product e * (fraction) do...

Honest Alice has error ≥ e with probability ≤ q

Alice makes few queries, regardless of Bob and Vera