View source for Fourier indices of an almost-periodic function

The real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041040/f0410401.png" /> in the Fourier series corresponding to the given almost-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041040/f0410402.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041040/f0410403.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041040/f0410404.png" /> are the Fourier coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041040/f0410405.png" /> (cf. [[Fourier coefficients of an almost-periodic function|Fourier coefficients of an almost-periodic function]]; [[Fourier series of an almost-periodic function|Fourier series of an almost-periodic function]]). The set of Fourier indices of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041040/f0410406.png" /> is called its spectrum. In contrast to the periodic case, the spectrum of an almost-periodic function can have finite limit points and can even be everywhere dense. Therefore, the behaviour of the Fourier series of an almost-periodic function depends in an essential way on the arithmetic structure of its spectrum.