Strong tractability for multivariate integration in a subspace of the Wiener algebra

Building upon recent work by the author, we prove that multivariate integration in the following subspace of the Wiener algebra over $[0,1{)}^d$ is strongly polynomially tractable: $$F_d:=\{f\in C([0,1{)}^d)|\mathrm{\parallel }f\mathrm{\parallel }:=\sum _{\mathit{k}\in {\mathbb{Z}}^d}\mathrm{\mid }\widehat{f}(\mathit{k})\mathrm{\mid }\max (\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mathit{k})),\underset{j\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mathit{k})}{\min }\log \mathrm{\mid }{k}_j\mathrm{\mid })<\mathrm{\infty }\},$$ with $\widehat{f}(\mathit{k})$ being the $\mathit{k}$-th Fourier coefficient of $f$, $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mathit{k}):=\{j\in \{1,\dots ,d\}\mid k_j\mathrm{\ne }0\}$, and $\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}:2^{\{1,\dots ,d\}}\to \{1,\dots ,d\}$ being defined by $$\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}(u):=\underset{j\in u}{\max }j-\underset{j\in u}{\min }j+1,$$ for non-empty subset $u\subseteq \{1,\dots ,d\}$ and $\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}(\mathrm{\varnothing }):=1$. Strong polynomial tractability is achieved by an explicit quasi-Monte Carlo rule using a multiset union of Korobov’s $p$-sets. We also show that, if we replace $\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\mathit{k}))$ with 1 for all $\mathit{k}\in {\mathbb{Z}}^d$ in the above definition of norm, multivariate integration is polynomially tractable but not strongly polynomially tractable.