iven a function $f : \{0,1\}^n \to \reals$, its {\em Fourier Entropy} is defined to be $-\sum_S \fcsq{f}{S} \log \fcsq{f}{S}$, where $\fhat$ denotes the Fourier transform of $f$. This quantity arises in a number of applications, especially in the study of Boolean functions. An outstanding open question is a conjecture ... more >>>

We study the complexity of approximating Boolean functions with DNFs and other depth-2 circuits, exploring two main directions: universal bounds on the approximability of all Boolean functions, and the approximability of the parity function.
In the first direction, our main positive results are the first non-trivial universal upper bounds on ... more >>>

We prove that, for all binary-input symmetric memoryless channels, polar codes enable reliable communication at rates within $\epsilon > 0$ of the Shannon capacity with a block length, construction complexity, and decoding complexity all bounded by a *polynomial* in $1/\epsilon$. Polar coding gives the *first known explicit construction* with rigorous ... more >>>