We consider the problem of testing if a given function $f : \mathbb{F}_2^n \rightarrow \mathbb{F}_2$ is close to any degree $d$ polynomial in $n$ variables, also known as the Reed-Muller testing problem. The Gowers norm is based on a natural $2^{d+1}$-query test for this property. Alon et al. [AKKLR05] rediscovered this test and showed that it accepts every degree $d$ polynomial with probability $1$, while it rejects functions that are $\Omega(1)$-far with probability $\Omega(1/(d 2^{d}))$. We give an asymptotically optimal analysis of this test, and show that it rejects functions that are (even only) $\Omega(2^{-d})$-far with $\Omega(1)$-probability (so the rejection probability is a universal constant independent of $d$ and $n$). This implies a tight relationship between the $(d+1)^{\rm{st}}$-Gowers norm of a function and its maximal correlation with degree $d$ polynomials, when the correlation is close to 1.
Our proof works by induction on $n$ and yields a new analysis of even the classical Blum-Luby-Rubinfeld [BLR93] linearity test, for the setting of functions mapping $\mathbb{F}_2^n$ to $\mathbb{F}_2$. The optimality follows from a tighter analysis of counterexamples to the ``inverse conjecture for the Gowers norm'' constructed by [GT09,LMS08].
Our result has several implications. First, it shows that the Gowers norm test is tolerant, in that it also accepts close codewords. Second, it improves the parameters of an XOR lemma for polynomials given by Viola and Wigderson [VW07]. Third, it implies a ``query hierarchy'' result for property testing of affine-invariant properties. That is, for every function $q(n)$, it gives an affine-invariant property that is testable with $O(q(n))$-queries, but not with $o(q(n))$-queries, complementing an analogous result of [GKNR09] for graph properties.
We consider the problem of testing if a given function
$f : \F_2^n \rightarrow \F_2$ is close to any degree $d$ polynomial
in $n$ variables, also known as the Reed-Muller testing problem.
Alon et al.~\cite{AKKLR} proposed and analyzed a natural
$2^{d+1}$-query test for this property and showed that it accepts
every degree $d$ polynomial with probability $1$, while rejecting
functions that are $\Omega(1)$-far with probability
$\Omega(1/(d 2^{d}))$.
We give an asymptotically optimal analysis of their test showing
that it rejects functions that are (even only) $\Omega(2^{-d})$-far
with $\Omega(1)$-probability (so the rejection probability
is a universal constant independent of $d$ and $n$).
Our proof works by induction on $n$, and yields a new analysis of
even the
classical Blum-Luby-Rubinfeld~\cite{BLR} linearity test, for the
setting of functions mapping $\F_2^n$ to
$\F_2$. The optimality follows from a tighter analysis of
counterexamples to the ``inverse conjecture
for the Gowers norm'' constructed by \cite{GT,LMS}.
Our result gives a new relationship between
the $(d+1)^{\rm{st}}$-Gowers norm of a function and its maximal
correlation with degree $d$ polynomials. For functions highly correlated
with degree $d$ polynomials, this relationship is asymptotically
optimal.
Our improved analysis of the \cite{AKKLR}-test also improves the
parameters of an XOR lemma for polynomials given by Viola and
Wigderson~\cite{VW}.
Finally, the optimality of our result also implies a ``query-hierarchy''
result for property testing of linear-invariant properties: For every
function $q(n)$, it gives a linear-invariant property that is testable
with $O(q(n))$-queries, but not with $o(q(n))$-queries, complementing
an analogous result of \cite{GKNR08} for graph properties.