Correction to: Almost periodic functions in terms of Bohr’s equivalence relation
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Correction to: Ramanujan J (2018) 46:245–267 https://doi.org/10.1007/s1113901799501
By taking into account this situation, in order to maintain the validity of Proposition 3 (not only for the case when it is possible to obtain an integral basis for the set of exponents) and some other subsequent results of our paper, the equivalence relation which is inspired by that of Bohr is now revised to adapt correctly the situation in the general case. In this way, Definition 3 of the above quoted paper is now modified in the following terms.
Definition 3′
Given \(\varLambda =\{\lambda _1,\lambda _2,\ldots ,\lambda _j,\ldots \}\) a set of exponents, consider \(A_1(p)\) and \(A_2(p)\) two exponential sums in the class \(\mathcal {S}_{\varLambda }\), say \(A_1(p)=\sum _{j\ge 1}a_je^{\lambda _jp}\) and \(A_2(p)=\sum _{j\ge 1}b_je^{\lambda _jp}.\) We will say that \(A_1\) is equivalent to \(A_2\) if for each integer value \(n\ge 1\), with \(n\le \sharp \varLambda \), it is satisfied \(a_{n}^*\sim b_{n}^*\), where \(a_{n}^*,b_{n}^*:\{\lambda _1,\lambda _2,\ldots ,\lambda _{n}\}\rightarrow \mathbb {C}\) are the functions given by \(a_{n}^*(\lambda _j):=a_j\) y \(b_{n}^*(\lambda _j):=b_j\), \(j=1,2,\ldots ,n\) and \(\sim \) is in Definition 1.
It is clear that the relation defined in the foregoing definition is an equivalence relation. That is, it is reflective, symmetric and transitive. By abuse of notation, we will use \(\sim \) for both equivalence relations introduced in Definitions 1 and 3\(^\prime \).
Analogously, Definition 5 must be rewritten in the following terms.
Definition 5′
Under the modification of Definition 3, Proposition 1 must now be rewritten in the following way.
Proposition 1′
Given \(\varLambda =\{\lambda _1,\lambda _2,\ldots ,\lambda _j,\ldots \}\) a set of exponents, consider \(A_1(p)\) and \(A_2(p)\) two exponential sums in the class \(\mathcal {S}_{\varLambda }\), say \(A_1(p)=\sum _{j\ge 1}a_je^{\lambda _jp}\) and \(A_2(p)=\sum _{j\ge 1}b_je^{\lambda _jp}.\) Fixed a basis \(G_{\varLambda }\) for \(\varLambda \), for each \(j\ge 1\) let \(\mathbf {r}_j\in \mathbb {R}^{\sharp G_{\varLambda }}\) be the vector of rational components verifying (2). Then \(A_1\sim A_2\) if and only if for any integer value \(n\ge 1\), with \(n\le \sharp \varLambda \), there exists \(\mathbf {x}_n=(x_{n,1},x_{n,2},\ldots ,x_{n,k},\ldots )\in \mathbb {R}^{\sharp G_{\varLambda }}\) such that \(b_j=a_j e^{\langle \mathbf {r}_j,\mathbf {x}_n\rangle i}\) for \(j=1,2,\ldots ,n\). Furthermore, if \(G_{\varLambda }\) is an integral basis for \(\varLambda \) then \(A_1\sim A_2\) if and only if there exists \(\mathbf {x}_0=(x_{0,1},x_{0,2},\ldots ,x_{0,k},\ldots )\in \mathbb {R}^{\sharp G_{\varLambda }}\) such that \(b_j=a_j e^{\langle \mathbf {r}_j,\mathbf {x}_0\rangle i}\) for every \(j\ge 1\).
Proof

In the proof of Lemma 1: “taking \(\mathbf {x}_0=\tau \mathbf {g}\)” must be changed by “taking \(\mathbf {x}_n=\tau \mathbf {g}\) for any integer value \(n\ge 1\)”.

In the proof of Lemma 2: “there exists \(\mathbf {x}_0\in \mathbb {R}^{\sharp \varLambda }\) such that \(b_j=a_j e^{\langle \mathbf {r}_j,\mathbf {x}_0\rangle i}\) for each \(j\ge 1\)” must be changed by “for any integer value \(n\ge 1\), with \(n\le \sharp \varLambda \), there exists \(\mathbf {x}_n\in \mathbb {R}^{\sharp \varLambda }\) such that \(b_j=a_j e^{\langle \mathbf {r}_j,\mathbf {x}_n\rangle i}\) for each \(j=1,\ldots ,n\)”.
 In the proof of Lemma 2: “define the sequence of trigonometric polynomialsmust be changed by “define the sequence of trigonometric polynomials$$\begin{aligned} Q_k(t):=\sum _{j\ge 1}p_{j,k}a_je^{\langle \mathbf {r}_j,\mathbf {x}_0\rangle i}e^{i\lambda _jt},\quad \ k=1,2,\ldots \hbox {''} \end{aligned}$$where, fixed k, we can take \(b_j=a_je^{\langle \mathbf {r}_j,\mathbf {x}_{n_k}\rangle i}\) with \(n_k\) the greatest integer value j such that \(p_{j,k}\ne 0\).”$$\begin{aligned} Q_k(t):=\sum _{j\ge 1}p_{j,k}b_je^{i\lambda _jt},\quad \ k=1,2,\ldots , \end{aligned}$$
 In the proof of Lemma 2: “However, note thatmust be changed by “However, note that$$\begin{aligned} M\{Q_{k_1}(t)Q_{k_2}(t)^2\}= & {} \displaystyle \sum _{j\ge 1}(p_{j,k_1}p_{j,k_2})^2\left e^{\langle \mathbf {r}_j,\mathbf {x}_0\rangle i}\right ^2a_j^2\\= & {} M\{P_{k_1}(t)P_{k_2}(t)^2\}\},\hbox {''} \end{aligned}$$where we suppose that \(n_{k_2}\ge n_{k_1}\).”$$\begin{aligned} M\{Q_{k_1}(t)Q_{k_2}(t)^2\}&=\displaystyle \sum _{j\ge 1}(p_{j,k_1}p_{j,k_2})^2\left e^{\langle \mathbf {r}_j,\mathbf {x}_{n_{k_2}}\rangle i} \right ^2a_j^2\\&=M\{P_{k_1}(t)P_{k_2}(t)^2\}, \end{aligned}$$

In the proof of Lemma 2: “we have \(f_2(t)\in AP(\mathbb {R},\mathbb {C})\). Moreover, \(\{Q_k(t)\}_{k\ge 1}\) also converges formally to the series \(\sum _{j\ge 1}a_j e^{\langle \mathbf {r}_j,\mathbf {x}_0\rangle i}e^{i\lambda _jt},\) which, by [1, p. 21], represents the Fourier series of \(f_2(t)\). Finally, by taking into account Definition 5 (in terms of Proposition 1) we have \(f_1\sim f_2\).” must be changed by “we have \(f_2(t)\in AP(\mathbb {R},\mathbb {C})\) and, by [1, p. 21], \(\sum _{j\ge 1}b_je^{i\lambda _jt}\) represents its Fourier series. Finally, by taking into account Definition 5 (in terms of Proposition 1\(^\prime \)) we have \(f_1\sim f_2\).”
 In the proof of Proposition 3: “Since \(f_1\sim f_l\) for each \(l=1,2,\ldots \), we deduce from Proposition 1 that there exists \(\mathbf {x}_l=(x_{l,1},x_{l,2},\ldots )\in \mathbb {R}^{\sharp G_\varLambda }\) such thatmust be changed by “Since \(f_1\sim f_l\) for each \(l=1,2,\ldots \), we deduce from Proposition 1\(^\prime \) that for any integer value \(n\ge 1\), with \(n\le \sharp \varLambda \), there exists \(\mathbf {x}_{l,n}=(x_{l,n,1},x_{l,n,2},\ldots )\in \mathbb {R}^{\sharp G_\varLambda }\) such that$$\begin{aligned} a_{l,j}=a_{1,j}e^{i\langle \mathbf {r}_j,\mathbf {x}_l\rangle },\ j=1,2\ldots ,\ \text{ with } \lambda _j\in \varLambda .\hbox {''} \end{aligned}$$(15)$$\begin{aligned} a_{l,j}=a_{1,j}e^{i\langle \mathbf {r}_j,\mathbf {x}_{l,n}\rangle },\ j=1,2\ldots ,n\ \text{ with } \lambda _j\in \varLambda .\hbox {''} \end{aligned}$$(15)
 In the proof of Proposition 3: “and, since (16) is satisfied for any \(k=1,2,\ldots \), we can also construct a vector \(\mathbf {x}_0\in \mathbb {R}^{\sharp G_{\varLambda }}\) such that, by taking into account remarks 1 and 2, verifiesmust be changed by “and, since (16) is satisfied for any \(k=1,2,\ldots \), we can construct, for any integer value \(n\ge 1\) with \(n\le \sharp \varLambda \), a vector \(\mathbf {x}_{0,n}\in \mathbb {R}^{\sharp G_{\varLambda }}\) such that, by taking into account remarks 1 and 2, verifies$$\begin{aligned} a_{j}=a_{1,j}e^{i\langle \mathbf {r}_j,\mathbf {x}_0\rangle },\ j=1,2\ldots , \text{ with } \lambda _j\in \varLambda .\hbox {''} \end{aligned}$$$$\begin{aligned} a_{j}=a_{1,j}e^{i\langle \mathbf {r}_j,\mathbf {x}_{0,n}\rangle },\ j=1,2\ldots ,n \text{ with } \lambda _j\in \varLambda .\hbox {''} \end{aligned}$$
Finally, the authors wish to thank Mattia Righetti for his quote to Bohr’s example given by the set of exponents \(\varLambda _0\) considered above [2, 3].
Notes
References
 1.Besicovitch, A.S.: Almost Periodic Functions. Dover, New York (1954)Google Scholar
 2.Bohr, H.: Zur Theorie der allgemeinen Dirichletschen Reihen. Math. Ann. (German) 79, 136–156 (1918)MathSciNetCrossRefzbMATHGoogle Scholar
 3.Righetti, M.: On Bohr’s equivalence theorem. J. Math. Anal. Appl. (Corrigendum Ibid.) 449, 939–940 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
 4.Sepulcre, J.M., Vidal, T.: Almost periodic functions in terms of Bohr’s equivalence relation. Ramanujan J. 46(1), 245–267 (2018)MathSciNetCrossRefzbMATHGoogle Scholar