Let $p$ be a fixed prime number, and $N$ be a large integer.
The 'Inverse Conjecture for the Gowers norm' states that if the "$d$-th Gowers norm" of a function $f:\F_p^N \to \F_p$ is non-negligible, that is larger than a constant independent of $N$, then $f$ can be non-trivially ... more >>>

We recover the first linear programming bound of McEliece, Rodemich, Rumsey, and Welch for binary error-correcting codes and designs via a covering argument. It is possible to show, interpreting the following notions appropriately, that if a code has a large distance, then its dual has a small covering radius and, ... more >>>

We define tests of boolean functions which
distinguish between linear (or quadratic) polynomials, and functions
which are very far, in an appropriate sense, from these
polynomials. The tests have optimal or nearly optimal trade-offs
between soundness and the number of queries.

In particular, we show that functions with small ... more >>>

We give a combinatorial proof of the result of Kahn, Kalai, and Linial, which states that every balanced boolean function on the n-dimensional boolean cube has a variable with influence of at least Omega(log(n)/n).The methods of the proof are then used to recover additional isoperimetric results for the cube, with ... more >>>

Gowers introduced, for d\geq 1, the notion of dimension-d uniformity U^d(f)
of a function f: G -> \C, where G is a finite abelian group and \C are the
complex numbers. Roughly speaking, if a function has small Gowers uniformity
of dimension d, then it ``looks random'' on ... more >>>

We consider the problem of approximating the entropy of a discrete distribution P on a domain of size q, given access to n independent samples from the distribution. It is known that n > q is necessary, in general, for a good additive estimate of the entropy. A problem of ... more >>>

For a family X of k-subsets of the set 1...n, let |X| be the cardinality of X and let Gamma(X, mu) be the expected maximum weight of a subset from X when the weights of 1...n are chosen independently at random from a symmetric probability distribution mu on R. We ... more >>>

We consider the problem of determining whether a given
function f : {0,1}^n -> {0,1} belongs to a certain class
of Boolean functions F or whether it is far from the class.
More precisely, given query access to the function f and given
a distance parameter epsilon, we would ... more >>>

We present improved algorithms for testing monotonicity of functions.
Namely, given the ability to query an unknown function $f$, where
$\Sigma$ and $\Xi$ are finite ordered sets, the test always accepts a
monotone $f$, and rejects $f$ with high probability if it is $\e$-far
from being monotone (i.e., every ... more >>>