Let
${{A}_{n}}\,=\,\{{{a}_{0}}+{{a}_{1}}z+\,.\,.\,.\,+\,{{a}_{n-1}}{{z}^{n-1}}\,:\,{{a}_{j}}\,\in \,\{0,1\}\}$
, whose elements are called zero-one polynomials and correspond naturally to the
${{2}^{n}}$
subsets of
$\left[ n \right]\,:=\,\{0,\,1,\,.\,.\,.\,,\,n-1\}$
. We also let
${{A}_{n,m\,}}\,=\,\{\alpha \left( z \right)\,\in \,{{A}_{n}}\,:\,\alpha \left( 1 \right)\,=\,m\}$
, whose elements correspond to the
$\left( _{m}^{n} \right)$
subsets of
$\left[ n \right]$
of size
$m$
, and let
${{B}_{n}}\,=\,{{A}_{n+1}}\,\backslash \,{{A}_{n}}$
, whose elements are the zero-one polynomials of degree exactly
$n$
.
Many researchers have studied norms of polynomials with restricted coefficients. Using
$\|\alpha {{\|}_{p}}$
to denote the usual
${{L}_{p}}$
norm of
$\alpha$
on the unit circle, one easily sees that
$\alpha \left( z \right)\,=\,{{a}_{0}}+{{a}_{1}}z+.\,.\,.+{{a}_{N}}{{z}^{N}}\,\in \,\mathbb{R}\left[ z \right]$
satisfies
$\|\alpha \|_{2}^{2}\,=\,{{c}_{0}}$
and
$\|\alpha \|_{4}^{4}\,=\,c_{0}^{2}\,+\,2\left( c_{1}^{2}\,+\,.\,.\,.\,+\,c_{N}^{2} \right)$
, where
${{c}_{k}}\,:=\,\sum _{j=0}^{N-k}\,{{a}_{j}}{{a}_{j+k}}\,\text{for}\,\text{0}\le \,\text{k}\,\le N$
.
If
$\alpha \left( z \right)\,\in \,{{A}_{n,m}}$
, say
$\alpha \left( z \right)\,=\,{{z}^{{{\text{ }\!\!\beta\!\!\text{ }}_{1}}}}\,+\,.\,.\,.\,+\,{{z}^{{{\text{ }\!\!\beta\!\!\text{ }}_{m}}}}$
where
${{\beta }_{1}}\,<\,.\,.\,.\,<\,{{\beta }_{m}}$
, then
${{c}_{k}}$
is the number of times
$k$
appears as a difference
${{\beta }_{i}}\,-\,{{\beta }_{j}}$
. The condition that
$\alpha \,\in \,{{A}_{n,m}}$
satisfies
${{c}_{k}}\,\in \,\{0,1\}$
for
$1\,\le \,k\,\le \,n\,-\,1$
is thus equivalent to the condition that
$\{{{\beta }_{1}},\,.\,.\,.\,,\,{{\beta }_{m}}\}$
is a Sidon set (meaning all differences of pairs of elements are distinct).
In this paper, we find the average of
$\left\| \alpha \right\|_{4}^{4}$
over
$\alpha \,\in \,{{A}_{n}}$
,
$\alpha \,\in \,{{B}_{n}}$
and
$\alpha \,\in \,{{A}_{n,m}}$
. We further show that our expression for the average of
$\left\| \alpha \right\|_{4}^{4}$
over
${{A}_{n,m}}$
yields a new proof of the known result: if
$m\,=\,o\left( {{n}^{1/4}} \right)$
and
$B\left( n,\,m \right)$
denotes the number of Sidon sets of size
$m$
in
$\left[ n \right]$
, then almost all subsets of
$\left[ n \right]$
of size
$m$
are Sidon, in the sense that
${{\lim }_{n\to \infty }}\,B\left( n,\,m \right)/\left( _{m}^{n} \right)\,=\,1$
.