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Let $n\ge 2$ be an integer and let $f$ be an entire function on $\mathbb C^n$. Let $A$ be a subset of $\mathbb R^n$ with positive $n$-dimensional Lebesgue measure. Then if $f$ vanishes at $A$, this implies that $f=0$.

Question. Is there an elementary proof of the above statement?

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    $\begingroup$ something like fixing a variable and considering the Taylor coefficients as analytic functions in the other $n-1$ variables and the projection of $A$ on the fixed variable has corresponding (real projection) non zero Lebesgue measure by Fubini, so all those coefficients must be identically zero etc $\endgroup$
    – Conrad
    Commented Feb 26 at 16:28
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    $\begingroup$ If $f$ is nonconstant, $f^{-1}(0)$ is a complex-analytic subvariety of dimension $n-1$. It's intersection with $R^n$ is a real-analytic subvariety of (real) dimension $\le n-1$... $\endgroup$
    – Moishe Kohan
    Commented Feb 26 at 16:30

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This follows immediatelly from the following paper:

The Zero Set of a Real Analytic Function

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  • $\begingroup$ Thx, it looks great. $\endgroup$
    – Bazin
    Commented Feb 26 at 16:47
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    $\begingroup$ @Bazin I needed this exact result for a project few years ago, recalled it:) You are welcome $\endgroup$
    – Nick S
    Commented Feb 26 at 17:41

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