Expected Norms of Zero-One Polynomials

Canadian mathematical bulletin

关键词: Hankel transform;    Laplace transform;    multiplier;    Calderón--Zygmund;   

DOI  :  10.4153/CMB-2008-050-2

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

Let $cA_n = ig{ a_0 + a_1 z + cdots + a_{n-1}z^{n-1} : a_j in {0, 1 } ig}$, whose elements are called emf{zero-one polynomials}and correspond naturally to the $2^n$ subsets of $[n] := { 0, 1,ldots, n-1 }$. We also let $cA_{n,m} = { alf(z) in cA_n :alf(1) = m }$, whose elements correspond to the ${n choose m}$subsets of~$[n]$ of size~$m$, and let $cB_n = cA_{n+1} setminuscA_n$, whose elements are the zero-one polynomials of degreeexactly~$n$.Many researchers have studied norms of polynomials with restrictedcoefficients. Using $orm{alf}_p$ to denote the usual $L_p$ normof~$alf$ on the unit circle, one easily sees that $alf(z) = a_0 +a_1 z + cdots + a_N z^N in R[z]$ satisfies $orm{alf}_2^2 = c_0$and $orm{alf}_4^4 = c_0^2 + 2(c_1^2 + cdots + c_N^2)$, where $c_k:= sum_{j=0}^{N-k} a_j a_{j+k}$ for $0 le k le N$.If $alf(z) in cA_{n,m}$, say $alf(z) = z^{eta_1} + cdots +z^{eta_m}$ where $eta_1 < cdots < eta_m$, then $c_k$ is thenumber of times $k$ appears as a difference $eta_i - eta_j$. Thecondition that $alf in cA_{n,m}$ satisfies $c_k in {0,1}$ for $1le k le n-1$ is thus equivalent to the condition that ${ eta_1,ldots, eta_m }$ is a emf{Sidon set} (meaning all differences ofpairs of elements are distinct).In this paper, we find the average of~$|alf|_4^4$ over $alf incA_n$, $alf in cB_n$, and $alf in cA_{n,m}$. We further showthat our expression for the average of~$|alf|_4^4$ over~$cA_{n,m}$yields a new proof of the known result: if $m = o(n^{1/4})$ and$B(n,m)$ denotes the number of Sidon sets of size~$m$ in~$[n]$, thenalmost all subsets of~$[n]$ of size~$m$ are Sidon, in the sense that$lim_{n o infty} B(n,m)/inom{n}{m} = 1$.

【 授权许可】

Unknown