A note on the maximal coefficients of squares of Newman polynomials

Abstract

In a recent paper [G. Yu, An upper bound for B 2 [g] sets, J. Number Theory 122 (1) (2007) 211-220] Gang Yu stated the following conjecture: Let {p i } ∞ i=1 be an arbitrary sequence of polynomials with increasing degrees and all coefficients in {0, 1}. If we denote by (#p i ) the number of non-zero coefficients of p i , and let M(p 2 i ) be the maximal coefficient of p 2 i , then Q := lim inf i→∞ deg(p i )M(p 2 i ) (#p i ) 2 1, ( * ) as long as (#p i ) = o(deg p i ), as i → ∞. We give an explicit example that shows why this last condition is necessary, and we investigate some open questions it suggests.

Heights of powers of Newman and Littlewood polynomials

“…Then deg(P ) = N (P ) = 3, H(P 2 ) = H(1+2x+x 2 +2x 3 +2x 4 +x 6 ) = 2, giving Q 2 (P ) = 8/9. Combined with Theorem 1 this implies the main result of [2]. Note that the polynomial 1 + x 2 + x 3 (which is reciprocal to 1 + x + x 3 ) already features as extremal in the following well-known problem: find f (n) = sup N (P )=n, P Newman inf |z|=1 |P (z)|.…”

“…(See Section 4 for some definitions and a discussion concerning B h [g] sets.) Berenhaut and Saidak [2] proved that the condition lim k→∞ N (P k )/deg(P k ) = 0 is indeed necessary in this conjecture. More precisely, they showed that there is a sequence of Newman polynomials for which the above limit is 8/9.…”

“…He conjectured (see Conjecture 2 in [22]) that this limit is always at least 1 if N (P k )/deg(P k ) → 0 as k → ∞. This conjecture, if proved, would give a sharp bound for the so-called B 2 [g] sets which generalize classical Sidon sets B 2 [1]. (See Section 4 for some definitions and a discussion concerning B h [g] sets.)…”

“…So we can restate the problem considered in [2] as follows: find the smallest possible limit lim inf k→∞ Q 2 (P k ) over all sequences of Newman polynomials P k , k = 1, 2, . .…”

“…The problem studied in [2] and [22] thus reduces to finding the infimum of the quantity Q 2 (P ), where P runs over all Newman polynomials. More generally, for higher powers, one may look for…”