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Let $\mu$ be a finite non negative singular measure on $\mathbf{R}^d$. I would like to know if there exists some result on the infimum of the absolute value of its Fourier Transform $$\hat{\mu}(t)=\displaystyle\int \mathrm{e}^{2i\pi t\cdot x} ~\mu(dx).$$

It is known that we don't necessarily have $\hat{\mu}(t)\rightarrow0$ for singular measure. But does one have like Lebesgue measure $$\mathrm{inf}~|\hat{\mu}(t)|=0~?$$

Edit : To be more precise, $\mu$ is continuous singular, so does not have a discret part (https://en.wikipedia.org/wiki/Lebesgue%27s_decomposition_theorem)

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    $\begingroup$ What about when $\mu$ is the Dirac delta? $\endgroup$
    – lcv
    Feb 24, 2022 at 22:59
  • $\begingroup$ Sorry, I wasn't specific enough. I wanted $\mu$ singular continuous (en.wikipedia.org/wiki/Lebesgue%27s_decomposition_theorem) $\endgroup$
    – Tiblodocus
    Feb 25, 2022 at 7:44
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    $\begingroup$ (1) If instead of $\inf |\hat\mu| = 0$ we require $\lim_{t \to \pm\infty} |\hat\mu(t)|=0$, we arrive at the notion of Rajchman measure. (2) By considering $\mu \star \check\mu$, where $\check\mu(A) = \mu(-A)$, with can reduce the problem to the case of symmetric measures. In this case one should be able to get some information about the support of $\mu$ — it is necessarily concentrated near certain arithmetic sequences — but I did not think about it much. $\endgroup$ Feb 25, 2022 at 9:05
  • $\begingroup$ Not directly relevant to your question, but it's worth noting that even if $\mu$ is discrete, the question of $\inf|\hat\mu|$ seems interesting: it will “often” turn out to be zero (e.g., if $\mu$ is the sum of two deltas in $\mathbb{R}$ concentrated on two reals with irrational ratio). Also, in this paper (esp. §5 and appendix B) we study $\inf\hat\mu$ (no absolute value!) in the case of (the discrete measure on) root systems in $\mathbb{R}^d$, which is equivalent to a result of Serre on characters of real Lie groups (proven in the appendix). $\endgroup$
    – Gro-Tsen
    Feb 25, 2022 at 9:23
  • $\begingroup$ (Sorry, I mean we need three reals with irrational ratio of differences for $\inf|\hat\mu|$ to be zero. Anyway, the point is that the question is clearly linked to problems of Diophantine approximation.) $\endgroup$
    – Gro-Tsen
    Feb 25, 2022 at 9:34

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Yes, that's true. By Wiener's lemma we have $\sum_{x\in \mathbb{R}^d} |\mu(\{x\})|^2 = \lim_{R\to \infty} \frac{1}{(2R)^d} \int_{[-R,R]^{d}} |\widehat{\mu}(t)|^2 dt$. If $\mu$ is continuous, then the left-hand side is $0$, so the infimum of $|\widehat{\mu}|$ has to be zero as well, since the right-hand side is bounded below by this infimum (squared).

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