Let \(\alpha\) be irrational and let \(f:\mathbb{R}/\mathbb{Z}\to\mathbb{R}\) be Riemann integrable with integral zero. Let \(f_ n(z)\) denote the Weyl sum \(f_ n(x):= \sum_{k=0}^{n-1} f(x+k\alpha\bmod 1)\), \(x\in\mathbb{R}/\mathbb{Z}\), \(n\in\mathbb{N}\). This type of sum appears in the theory of uniform distribution of sequences modulo one and in ergodic theory. Various authors have presented conditions for \(\alpha\) and \(f\) such that the Weyl sums \(f_ n(x)\) are bounded in \(n\). Define the following set of irrational numbers: \({\mathcal B}(f):=\{\alpha\) irrational: \(\sup_{n\in\mathbb{N}} \| f_ n\|_{L^ \infty(\lambda)}<\infty\}\). In this paper it is shown how the condition ``\(\alpha\in{\mathcal B}(f)\)'' is related to conditions on the Fourier coefficients of the function \(f\). Counter-examples demonstrate that, in general, these conditions are not equivalent. In the second theorem of this paper an inequality between the sums \(\sum_{k\neq 0} (| k|^ s\| k\alpha\|)^{-t}\), \(s\geq 2\), \(t\geq 1\), \(s\) and \(t\) real, and \(\sum_ i(a_{i+1}/q_ i)^ t\), where \(a_ 1,a_ 2,\dots\) are the partial quotients and \(q_ 0,q_ 1,\dots\) the denominators of the convergents to \(\alpha\) is proved. It follows as a corollary that growth conditions on the Fourier coefficients of \(f\) imply sufficient (diophantine) conditions such that \(\alpha\) belongs to \({\mathcal B}(f)\). This technique allows shorter proofs and improvements of several known results. The relevant coboundary theorems of ergodic theory are discussed in an appendix.