Three sine functions

All zeros of the function Sin[a x] + Sin[b x]

Sin[a x] + Sin[b x] lie on the real axis. But for Sin[a x] + Sin[b x] + Sin[c x]

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Sin[a x] + Sin[b x] + Sin[c x], there are usually zeros off the real axis (even say for a = 1

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a = 1, b = 3/2

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b = 3/2, c = 5/3

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c = 5/3), as shown in the pictures below.

If a, b and c are rational, Sin[a x] + Sin[b x] + Sin[c x]

Sin[a x] + Sin[b x] + Sin[c x] is periodic with period 2 π /GCD[a, b, c]

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2 π /GCD[a, b, c], and there are a limited number of different spacings between zeros. But in a case like Sin[x] + Sin[√2 x] + Sin[√3 x]

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Sin[x]+Sin[\!\(\*SqrtBox[\(2\)]\)x]+Sin[\!\(\*SqrtBox[\(3\)]\)x] there is a continuous distribution of spacings between zeros, as shown on a logarithmic scale below. (For 0 < x < 10⁶

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0\[RawLess]x\[RawLess]\!\(\*SuperscriptBox[\(10\),\(6\)]\) there are a total of 448,494 zeros, with maximum spacing ≃4.6

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≃4.6 and minimum spacing ≃0.013

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≃0.013.)