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Ben Green
Ben Green
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The existing answers list some important situations where Poisson Summation plays a role, the application to proving the functional equation of $\theta$ and hence of $\zeta$ being my personal favourite. My best answer to Tim's question as he actually asked it might be: why not have it in mind to try using it whenever you have a discrete sum that you are having trouble estimating, especially if you fancy your chances of understanding the Fourier transform of the summands. You'll end up with a different sum and it might be a lot easier to understand, and you might even be able to approximate your first sum by an integral (the term $\hat{f}(0)$ in the Poisson summation formula).

To explain a little more with an example, there's a whole theory concerned with the estimation of exponential sums $\sum_{n \leq N} e^{2\pi i \phi(n)}$. There are two processes called A and B that can be used to turn a sum like this into something you might be better positioned to understand. Process A is basically Weyl/van der Corput differencing (CuachyCauchy-Schwarz) and process B is essentially Poisson summation. It's not a very straightforward task to put together a theory of how these processes interact, and how they may best be combined to estimate your sum, and in fact this is in general something of an art. The 10 lectures book by Montgomery contains a nice exposition, and there's a whole LMS lecture note volume by Graham and Kolesnik if you want more details.

I want to share a perhaps slightly obscure paper of Roberts and Sargos (Three-dimensional exponential sums with monomials, Journal fur die reine und angewandte Mathematik (Crelle) 591), in which they use Poisson Summation in the form of Process B mentioned above arbitrarily many times to establish the following rather simple-to-state result: the number of quadruples $x_1,x_2,x_3,x_4$ in $[X, 2X)$ with

$$|1/x_1 + 1/x_2 - 1/x_3 - 1/x_4| \leq 1/X^3$$

is $X^{2 + o(1)}$. In other words, the quantities $1/a + 1/b$ tend to avoid one another to pretty much the same extent as random numbers of the same size. Very very roughly speaking (I don't really understand the argument in depth) the proof involves looking at exponential sums $\sum_x e^{2\pi i m/x}$, and it is these that are transformed repeatedly using Poisson summation followed by other modifications (it being reasonably pointless to try and apply Poisson sum twice in succession).

The existing answers list some important situations where Poisson Summation plays a role, the application to proving the functional equation of $\theta$ and hence of $\zeta$ being my personal favourite. My best answer to Tim's question as he actually asked it might be: why not have it in mind to try using it whenever you have a discrete sum that you are having trouble estimating, especially if you fancy your chances of understanding the Fourier transform of the summands. You'll end up with a different sum and it might be a lot easier to understand, and you might even be able to approximate your first sum by an integral (the term $\hat{f}(0)$ in the Poisson summation formula).

To explain a little more with an example, there's a whole theory concerned with the estimation of exponential sums $\sum_{n \leq N} e^{2\pi i \phi(n)}$. There are two processes called A and B that can be used to turn a sum like this into something you might be better positioned to understand. Process A is basically Weyl/van der Corput differencing (Cuachy-Schwarz) and process B is essentially Poisson summation. It's not a very straightforward task to put together a theory of how these processes interact, and how they may best be combined to estimate your sum, and in fact this is in general something of an art. The 10 lectures book by Montgomery contains a nice exposition, and there's a whole LMS lecture note volume by Graham and Kolesnik if you want more details.

I want to share a perhaps slightly obscure paper of Roberts and Sargos (Three-dimensional exponential sums with monomials, Journal fur die reine und angewandte Mathematik (Crelle) 591), in which they use Poisson Summation in the form of Process B mentioned above arbitrarily many times to establish the following rather simple-to-state result: the number of quadruples $x_1,x_2,x_3,x_4$ in $[X, 2X)$ with

$$|1/x_1 + 1/x_2 - 1/x_3 - 1/x_4| \leq 1/X^3$$

is $X^{2 + o(1)}$. In other words, the quantities $1/a + 1/b$ tend to avoid one another to pretty much the same extent as random numbers of the same size. Very very roughly speaking (I don't really understand the argument in depth) the proof involves looking at exponential sums $\sum_x e^{2\pi i m/x}$, and it is these that are transformed repeatedly using Poisson summation followed by other modifications (it being reasonably pointless to try and apply Poisson sum twice in succession).

The existing answers list some important situations where Poisson Summation plays a role, the application to proving the functional equation of $\theta$ and hence of $\zeta$ being my personal favourite. My best answer to Tim's question as he actually asked it might be: why not have it in mind to try using it whenever you have a discrete sum that you are having trouble estimating, especially if you fancy your chances of understanding the Fourier transform of the summands. You'll end up with a different sum and it might be a lot easier to understand, and you might even be able to approximate your first sum by an integral (the term $\hat{f}(0)$ in the Poisson summation formula).

To explain a little more with an example, there's a whole theory concerned with the estimation of exponential sums $\sum_{n \leq N} e^{2\pi i \phi(n)}$. There are two processes called A and B that can be used to turn a sum like this into something you might be better positioned to understand. Process A is basically Weyl/van der Corput differencing (Cauchy-Schwarz) and process B is essentially Poisson summation. It's not a very straightforward task to put together a theory of how these processes interact, and how they may best be combined to estimate your sum, and in fact this is in general something of an art. The 10 lectures book by Montgomery contains a nice exposition, and there's a whole LMS lecture note volume by Graham and Kolesnik if you want more details.

I want to share a perhaps slightly obscure paper of Roberts and Sargos (Three-dimensional exponential sums with monomials, Journal fur die reine und angewandte Mathematik (Crelle) 591), in which they use Poisson Summation in the form of Process B mentioned above arbitrarily many times to establish the following rather simple-to-state result: the number of quadruples $x_1,x_2,x_3,x_4$ in $[X, 2X)$ with

$$|1/x_1 + 1/x_2 - 1/x_3 - 1/x_4| \leq 1/X^3$$

is $X^{2 + o(1)}$. In other words, the quantities $1/a + 1/b$ tend to avoid one another to pretty much the same extent as random numbers of the same size. Very very roughly speaking (I don't really understand the argument in depth) the proof involves looking at exponential sums $\sum_x e^{2\pi i m/x}$, and it is these that are transformed repeatedly using Poisson summation followed by other modifications (it being reasonably pointless to try and apply Poisson sum twice in succession).

Source Link
Ben Green
Ben Green
  • 4.8k
  • 2
  • 37
  • 35

The existing answers list some important situations where Poisson Summation plays a role, the application to proving the functional equation of $\theta$ and hence of $\zeta$ being my personal favourite. My best answer to Tim's question as he actually asked it might be: why not have it in mind to try using it whenever you have a discrete sum that you are having trouble estimating, especially if you fancy your chances of understanding the Fourier transform of the summands. You'll end up with a different sum and it might be a lot easier to understand, and you might even be able to approximate your first sum by an integral (the term $\hat{f}(0)$ in the Poisson summation formula).

To explain a little more with an example, there's a whole theory concerned with the estimation of exponential sums $\sum_{n \leq N} e^{2\pi i \phi(n)}$. There are two processes called A and B that can be used to turn a sum like this into something you might be better positioned to understand. Process A is basically Weyl/van der Corput differencing (Cuachy-Schwarz) and process B is essentially Poisson summation. It's not a very straightforward task to put together a theory of how these processes interact, and how they may best be combined to estimate your sum, and in fact this is in general something of an art. The 10 lectures book by Montgomery contains a nice exposition, and there's a whole LMS lecture note volume by Graham and Kolesnik if you want more details.

I want to share a perhaps slightly obscure paper of Roberts and Sargos (Three-dimensional exponential sums with monomials, Journal fur die reine und angewandte Mathematik (Crelle) 591), in which they use Poisson Summation in the form of Process B mentioned above arbitrarily many times to establish the following rather simple-to-state result: the number of quadruples $x_1,x_2,x_3,x_4$ in $[X, 2X)$ with

$$|1/x_1 + 1/x_2 - 1/x_3 - 1/x_4| \leq 1/X^3$$

is $X^{2 + o(1)}$. In other words, the quantities $1/a + 1/b$ tend to avoid one another to pretty much the same extent as random numbers of the same size. Very very roughly speaking (I don't really understand the argument in depth) the proof involves looking at exponential sums $\sum_x e^{2\pi i m/x}$, and it is these that are transformed repeatedly using Poisson summation followed by other modifications (it being reasonably pointless to try and apply Poisson sum twice in succession).