A note on the maximal coefficients of squares of Newman polynomials

In a recent paper [12] Gang Yu stated the following conjecture: Let {pi}i=1 be an arbitrary sequence of polynomials with increasing degrees and all coefficients in {0, 1}. If we denote by (#pi) the number of non-zero coefficients of pi, and let M(pi ) be the maximal coefficient of pi , then Q := lim inf i→∞ deg(pi) M(pi ) (#pi) ≥ 1, (∗) as long as (#pi) = o(deg pi), as i → ∞. We give an explicit example that shows why this last condition is necessary, and we investigate some questions it suggests.