In this paper we study the fractional block sensitivityof Boolean functions. Recently, Tal (ITCS, 2013) and
Gilmer, Saks, and Srinivasan (CCC, 2013) independently introduced this complexity measure, denoted by $fbs(f)$, and showed
that it is equal (up to a constant factor) to the randomized certificate complexity, denoted by $RC(f)$, which ... more >>>

In this paper, two structural results concerning low degree polynomials over the field $\mathbb{F}_2$ are given. The first states that for any degree d polynomial f in n variables, there exists a subspace of $\mathbb{F}_2^n$ with dimension $\Omega(n^{1/(d-1)})$ on which f is constant. This result is shown to be tight. ... more >>>

We give a function $h:\{0,1\}^n\to\{0,1\}$ such that every deMorgan formula of size $n^{3-o(1)}/r^2$ agrees with $h$ on at most a fraction of $\frac{1}{2}+2^{-\Omega(r)}$ of the inputs. This improves the previous average-case lower bound of Komargodski and Raz (STOC, 2013).

Our technical contributions include a theorem that shows that the ``expected ... more >>>

In this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of $f$ is $\|\hat{f}\|_1=\sum_{\alpha}|\hat{f}(\alpha)|$). Specifically, we prove the following results for functions $f:\{0,1\}^n\to \{0,1\}$ with $\|\hat{f}\|_1=A$.

1. There is a subspace $V$ of co-dimension at most $A^2$ such that $f|_V$ is constant.

2. ... more >>>

For Boolean functions $f:\{0,1\}^n \to \{0,1\}$ and $g:\{0,1\}^m \to \{0,1\}$, the function composition of $f$ and $g$ denoted by $f\circ g : \{0,1\}^{nm} \to \{0,1\}$ is the value of $f$ on $n$ inputs, each of them is the calculation of $g$ on a distinct set of $m$ Boolean variables. Motivated ... more >>>

We study the following problem raised by von zur Gathen and Roche:

What is the minimal degree of a nonconstant polynomial $f:\{0,\ldots,n\}\to\{0,\ldots,m\}$?

Clearly, when $m=n$ the function $f(x)=x$ has degree $1$. We prove that when $m=n-1$ (i.e. the point $\{n\}$ is not in the range), it must be the case ... more >>>

In this paper we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any non-linear symmetric Boolean function.
Specifically, we prove that for every non-linear and symmetric $f:\{0,1\}^{k} \to \{0,1\}$ there exists a set $\emptyset\neq S\subset[k]$ such that $|S|=O(\Gamma(k)+\sqrt{k})$, and $\hat{f}(S) \neq 0$, where ... more >>>