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I believe this question will best be served with the following example: What can we say about the minimum of $$f(x)=\sin(x)+\sin(x*\pi)?$$

I would conjecture that if we defined $x_n$ as the $nth$ local minimum of $f(x)$, then $$\liminf_{n\to\inf}x_n=-2.$$

More generally, if we define $$f(x)=\sum_{n=1}^r g_n(x)$$ where $r$ is some large integer and $g_n(x)$ are continuous, incommensurable, periodic functions, can we say $\liminf_{n\to\inf}x_n$ is equal to the sum of the minimums of $g_n(x)$? Links to papers or just terms to research appreciated as I'd like to learn more than the question posted above.

QC_QAOA
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1 Answers1

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For the general case, if $p_j$ are the periods, what you want is that the image of the ray from the origin in the direction $[1/p_1,\ldots,1/p_n]$ under the quotient map $\mathbb R^n \to \mathbb R^n/\mathbb Z^n$ is dense. This is the case if there do not exist nontrivial relations $\sum_{i=1}^n a_i/p_i = 0$ with $a_i$ integers, not all $0$.

Robert Israel
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