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This is almost certainly routine to an analyst, so forgive me in advance.

Let $\alpha_i\in \mathbb{R}$. Consider the functional $$\varphi: L^1[0.9A,A]\to \mathbb{C}$$ via $$f\mapsto \sum_i \hat{f}(\alpha_i),$$ where by '$\hat{f}$' I mean the Fourier transform of $f$ regarded as a function on $\mathbb{R}$ (via extension by zero).

What can one say about the norm of $\varphi$?

I am wondering about the case where all $|\alpha_i| = O(1)$, $A$ is large, and the number of $\alpha_i$ is also large (comparable to $A$). As the title suggests, I'm hoping for a lower bound on $||\varphi||$. The thing that seemed most natural to me was to write this as integrating against a sum of exponentials (say $g(t)$, so that $||g||_\infty = ||\varphi||$) and then calculate the $L^2$ norm of $g$, but I can't bound the off-diagonal terms well enough.

The idea is that if, for some $t$ in the interval, $g(t)$ is very small, then some small-order derivative of $g$ should be bounded below so that $g((1\pm \delta)t)\gg_\delta A^{-O(1)}$ maybe. One could get a lower bound on some derivative via the nonvanishing of a Vandermonde determinant, but I think that would require too many derivatives.

Perhaps I am mistaken and there is an example with $||\varphi||\ll \exp(-c A)$?

Anyway, thanks much!

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  • $\begingroup$ If we let the number go to $\infty$ and $A$ is fixed we have a good lower bound, and if $A$ is fixed and the number goes to $\infty$ then we also have a good lower bound. This makes it seem like there is a good lower bound $\endgroup$
    – Will Sawin
    Sep 23 '15 at 3:38
  • $\begingroup$ Actually expressing it in terms of the $A$ variable is kind of strange. The really relevant variables are $O(1)/A$ (the maximum frequency after normalizing since $A=1$) and the number of $\alpha_i$. $\endgroup$
    – Will Sawin
    Sep 23 '15 at 3:53
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I think I can prove a polynomial bound using complex analysis. This really seems like it shouldn't work but it seems to.

The bound is that for all $\alpha_1, \dots, \alpha_n \in \mathbb R$

$$ \sup_{t \in [A,B]}\left| \sum_{i=1}^n e( \alpha_i t) \right| \geq n^{1-\frac{\pi}{4 \arctan {\sqrt{\frac{B}{A}}}-\pi}}$$

So in the special case $[.9A,A]$ this is $n^{-28.8313 \dots}$.

We may freely translate the $\alpha_i$, therefore, we assume $\min \alpha_i=0$. Form the holomorphic function

$$f(z) = \sum_{i=1}^n e(\alpha_i z)$$

We have the following estimates:

In the whole upper half plane, $|f(z)| \leq n$, as all the $\alpha_i$ are nonnegative so $e(\alpha_i z) \leq 1$.

On the imaginary line, $|f(z)| \geq 1$, as some $\alpha_i$ is zero giving $e(\alpha_i z) =1$ and the rest of the terms are positive reals.

Assume that for $z$ on the real line in the interval $[A,B]$, $|f(z)|<C$. (Then the same holds for $[-B,-A]$, which helps to get a better constant but isn't essential).

$f$ is holomorphic so $\log | f(z)|$ is subharmonic. Hence by the mean value theorem for subharmonic functions and the various bounds we stated on $f$:

$$ 0 \leq \log |f(it)| \leq\frac{t}{2\pi} \int_{-\infty}^{\infty} \frac{\log |f(s)|}{t^2+s^2}ds \leq \frac{t}{2\pi} \int_{-\infty}^{\infty} \frac{\log n}{t^2+s^2}ds+ 2\frac{t}{2\pi} \int_{A}^{B} \frac{\log C - \log n}{t^2+s^2}ds$$

$$ = \log n + \frac{\theta}{\pi} (\log C - \log n)$$

where $\theta$ is the hyperbolic angle between $A$ and $B$ from $it$. So we have:

$$0 \leq \log n + \frac{\theta}{\pi} (\log C - \log n)$$

$$ \frac{\theta}{\pi} (\log n - \log C) \leq \log n$$

$$ \log C \geq (1- \frac{\pi}{\theta}) \log n$$

$$C \geq n^{1-\pi/\theta}$$

as desired.

The optimal value of $\theta$ is a hyperbolic trigonometry problem which I may or may not have gotten the right answer to. Anyways it's some explicit formula.

Tao's example starting with, $\alpha_1=0$, $\alpha_2= \frac{1}{A+B}$ has an upper bound of the form $n^{\frac{\log \left|1- e\left(\frac{A}{A+B}\right)\right|}{\log 2}}$. My exponent is far from this as $A$ goes to $B$ so sharpness is an interesting question.

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  • $\begingroup$ Wait --- do you mean +, rather than -, on all the (log C - log n) terms? Otherwise isn't the conclusion log C <= (1 + \theta/\pi) log n? $\endgroup$
    – alpoge
    Jan 19 '16 at 14:46
  • $\begingroup$ Also why the 2 in front of the \int_A^B? $\endgroup$
    – alpoge
    Jan 19 '16 at 14:50
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    $\begingroup$ @alpoge The $2$ is because there is also the interval $-B$ to $-A$. There's something messed up with the signs. Let me fix it. $\endgroup$
    – Will Sawin
    Jan 19 '16 at 14:55
  • $\begingroup$ Also I think there should be a t in front of the Poisson integral, but obviously it doesn't matter cause you just take t = 1 anyway. This is amazing!!!!! $\endgroup$
    – alpoge
    Jan 19 '16 at 14:56
  • $\begingroup$ @alpoge The signs are now right. Well technically to get the best constant taking $t= \sqrt{ab}$ so it matters some. $\endgroup$
    – Will Sawin
    Jan 19 '16 at 14:57
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One can indeed make $\|\phi\|$ exponentially small in $A$ (though the example I have requires an exponentially large number of frequencies $\alpha_i$).

We use the dual formulation, that is we find a sum of exponentials $g$ whose $L^\infty$ norm is small on $[0.9A, A]$. Firstly, if we consider the sum of just two exponentials $g_0(t) = 1 + e^{\pi i/A}$, then there is significant cancellation on $[0.9A,A]$, in particular $\|g_0\|_{L^\infty} \leq c < 1$ for some absolute constant $c<1$. If we then raise $g_0$ to the power $A$ (let's say $A$ is an integer) then we get a sum of exponentials $g_0^A$ whose frequencies are still bounded by 1, and whose sup norm decays exponentially fast in A.

I would expect exponential decay to be best possible, in particular the Vandermonde determinant approach probably gives such a bound. EDIT: it appears that one has the more general bound $\|\mu\|_1 \ll \exp(CA) \| \hat \mu \|_{L^\infty([0.9A,A])}$ for any nonnegative measure supported on $[-1,1]$, by using a variant of the real-variable proof of Hardy's uncertainty principle (see e.g. https://terrytao.wordpress.com/2009/02/18/hardys-uncertainty-principle/ ) to use $L^\infty$ control of $\hat \mu$ on $[0.9A,A]$ and a "doubling bound" to control $\hat \mu(0) = \|\mu\|_1$ with an exponential loss in constants, after first normalising $\|\mu\|_1=1$ to control error terms.

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  • $\begingroup$ Cool! The regime I'm interested in has the number of frequencies bounded by $\eps A$, $\eps > 0$ small. In this case your example gives polynomial decay, which is acceptable to me. (I realize now that I left out a factor of $A^{-O(1)}$ in my speculation on the lower bound above.) Did you mean you also expect an exponential lower bound in this case to be best possible? Thanks much again! [I'm not sure if I did this "@..." construction correctly, hopefully it works!] $\endgroup$
    – alpoge
    Sep 23 '15 at 17:05
  • $\begingroup$ By the way, I should mention that this Vandermonde approach doesn't use the fact that all the coefficients are $+1$, just that they are constant. Hence it would need knowledge about the gaps between the frequencies, because of examples like $1-e^{2\pi i \alpha t}$ with $\alpha = o(1/A)$ (or its powers, as above). What prevents such an example here is that if ${\alpha t}$ (the frac. part) is very close to $1/2$, then automatically $\alpha \gg 1/A$ and so by changing $t$ by a multiplicative constant we can move the fractional part by an additive constant. But I can't seem to make this precise :( $\endgroup$
    – alpoge
    Sep 23 '15 at 17:21

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