Coefficients of squares of Newman polynomials

We show that there are polynomials $p_N$ of arbitrarily large degree $N$, with coefficients equal to 0 or 1 (Newman polynomials), such that $$ \liminf_{N \to \infty} N \Linf{p_N^2} \bigl / p_N^2(1) < 1, $$ where $\Linf{q}$ denotes the maximum coefficient of the polynomial $q$ and which, at the same time, are sparse: $p_N(1)/N \to 0$. This disproves a conjecture of Yu \cite{yu}. We build on some previous results of Berenhaut and Saidak \cite{berenhaut-saidak} and Dubickas \cite{dubickas} whose examples lacked the sparsity. This sparsity we create from these examples by randomization.