A note on the maximal coefficients of squares of Newman polynomials

In a recent paper [G. Yu, An upper bound for B2[g]B2[g] sets, J. Number Theory 122 (1) (2007) 211–220] 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}{0,1}. If we denote by (#pi)(#pi) the number of non-zero coefficients of pipi, and let M(pi2) be the maximal coefficient of pi2, thenequation(∗)Q:=lim infi→∞deg(pi)M(pi2)(#pi)2⩾1, as long as (#pi)=o(degpi)(#pi)=o(degpi), as i→∞i→∞. We give an explicit example that shows why this last condition is necessary, and we investigate some open questions it suggests.