Let \(s\gt 0\) be an integer and \(Y_1,\ldots ,Y_s\) be independent real functions on a probability space \((\Omega ,\mu )\), each measurable up to a null set and valued in \([-1,1]\) almost surely, with common mean \(\mathbb E Y_i=m\). Write \(\Pr (A)=\mu ^*(A)\) for outer probability. For every real \(\varepsilon \ge 0\), \(\Pr [s\varepsilon \le \sum _iY_i-sm]\le \exp (-s\varepsilon ^2/2)\).

Let \(s\gt 0\) be an integer and \(Y_1,\ldots ,Y_s\) be independent real functions on a probability space \((\Omega ,\mu )\), each measurable up to a null set and valued in \([-1,1]\) almost surely, with common mean \(\mathbb E Y_i=m\). Write \(\Pr (A)=\mu ^*(A)\) for outer probability. For every real \(\varepsilon \ge 0\), \(\Pr [s\varepsilon \le |\sum _iY_i-sm|]\le 2\exp (-s\varepsilon ^2/2)\).

For every positive integer \(n\), \(2^{n+1}e^{-2n}\lt 1\).

Let \(n,s\ge 0\) be integers and \(L,\delta \gt 0\) be real numbers. If \(4nL^2/\delta ^2\le s\), then \(2n\le s(\delta /L)^2/2\).

Let \(n\gt 0\) and \(s\ge 0\) be integers and \(L,\delta \gt 0\) be real numbers with \(4nL^2/\delta ^2\le s\). Then \(2^n\bigl(2\exp (-s(\delta /L)^2/2)\bigr)\lt 1\).

Let \((\Omega ,\mu )\) be a probability space, let \(J\) be a finite index set, and let \((A_j)_{j\in J}\) be arbitrary subsets of \(\Omega \). If a real number \(B\) satisfies \(\mu ^*(A_j)\le B\) for every \(j\) and \(|J|B\lt 1\), then some \(\omega \in \Omega \) belongs to none of the \(A_j\), where \(\mu ^*\) denotes the outer measure of \(\mu \).

Let \(n\ge 0\) and give \(\{ 0,1\} ^n\) uniform probability; write \(\mathbb E\) for expectation. Put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for \(S\subseteq [n]\) and real cube functions \(h\). Write \(\operatorname {sgn}_+(t)=1\) for \(t\ge 0\) and \(-1\) for \(t\lt 0\). For a real cube function \(f\) with \(L=\sum _{S\subseteq [n]}|\widehat f(S)|\gt 0\), there exists a probability distribution \(\pi \) on the subsets of \([n]\) such that \(\mathbb E_{S\sim \pi }[\operatorname {sgn}_+(\widehat f(S))\chi _S(x)]=f(x)/L\) for every cube point \(x\).

Let \(n\ge 0\) and give \(\{ 0,1\} ^n\) uniform probability; write \(\mathbb E\) for expectation. Put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for \(S\subseteq [n]\) and real cube functions \(h\). If a real cube function \(f\) is not identically zero, then its spectral one-norm \(\sum _{S\subseteq [n]}|\widehat f(S)|\) is strictly positive.

Let \(n\ge 0\) and give \(\{ 0,1\} ^n\) uniform probability; write \(\mathbb E\) for expectation. Put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for \(S\subseteq [n]\) and real cube functions \(h\). Write \(\operatorname {sgn}_+(t)=1\) for \(t\ge 0\) and \(-1\) for \(t\lt 0\). Let \(n\gt 0\), let \(f\) be a real cube function not identically zero, put \(L=\sum _S|\widehat f(S)|\), and let \(\delta \gt 0\) and an integer \(s\ge 0\) satisfy \(4nL^2/\delta ^2\le s\). There exists a sequence of subsets \(T_1,\ldots ,T_s\subseteq [n]\), with repetitions allowed, such that \(|f(x)-(L/s)\sum _{j=1}^s\operatorname {sgn}_+(\widehat f(T_j))\chi _{T_j}(x)|\lt \delta \) for every \(x\in \{ 0,1\} ^n\).