This answer is based on the handwritten notes [1] that are linked in the question, but I wrote it in my own words and used slightly different conventions: the outcomes are labelled $0,1$ instead of $pm 1$, and the outcomes in the same-angles case are always equal to each other instead of always opposite. (This can be arranged simply by changing the reference-direction with respect to which one of the angles is defined.)

The phenomenon

Suppose we have a device with one button. Each time we press the button, the device shoots out a pair of particles, $A$ and $B$, travelling in opposite directions. After the particles are far away from each other, we can measure an observable of particle $A$ along an axis with orientation $alpha$, and we measure an observable of particle $B$ along an axis with orientation $beta$. The particles and the observables are chosen so that each measurement has only two possible outcomes, which we can label $1$ and $0$.

Quantum theory predicts (and experimentes confirm) that we can design the device to produce a special state with the following properties: First, when the angles are equal, the outcomes are either both $1$ or both $0$. Second, no matter what angles $alpha,beta$ we choose, the fraction of trials in which both outcomes are $1$ approaches $$ p(alpha-beta) := frac{1+cos(alpha-beta)}{4} tag{1} $$ if the number of trials is large enough. Loosely speaking, (1) is the "probability" that both outcomes are $1$ when the angles are $alpha,beta$.

An impossible project

The project is to write a pair of computer programs, $A$ and $B$, that satisfy the following requirements.

• Program $A$ runs on computer $A$, and program $B$ runs on computer $B$. The programs may have as many common features as we want, but once the experiment begins, they are not allowed to communicate with each other.

• Each program takes a sequence of $N$ angles as input and returns a sequence of $N$ binary digits. Let $A_n(alpha)$ and $B_n(beta)$ denote the binary digit that will be returned by programs $A$ and $B$, respectively, if the angles used in the $n$th trial are $alpha$ and $beta$ (for $A$ and $B$, respectively).

• The programs should be designed so that $$ A_n(theta)=B_n(theta) tag{2a} $$ and $$ lim_{Ntoinfty}frac{1}{N}sum_n A_n(alpha)B_n(beta) = p(alpha-beta) tag{2b} $$ where $p(alpha,beta)$ is the function defined in equation (1).

This is impossible. Bell gave one proof, which is reviewed in [2]. I'll review Gull's proof instead because that's what the question is about.

Gull's proof that the project is impossible

Use (2a) in (2b) to get $$ lim_{Ntoinfty}frac{1}{N}sum_n A_n(alpha)A_n(beta) = p(alpha-beta) tag{3} $$ Let $tilde A_n$ denote the Fourier transform of $A_n$, so that $$ A_n(alpha) = sum_a e^{iaalpha}tilde A_n(a) tag{4} $$ where the sum is over all integers $ainmathbb{Z}$. Insert this expression for $A_n$ into equation (3) to get the equivalent requirement $$ lim_{Ntoinfty}frac{1}{N}sum_{n,a,b} e^{iaalpha}e^{ibbeta}tilde A_n(a)tilde A_n(b) = p(alpha-beta). tag{5} $$ Rewrite the angles as $$ alpha = phi+theta hskip2cm beta = phi-theta tag{6} $$ to get $$ lim_{Ntoinfty}frac{1}{N}sum_{n,a,b} e^{i(a+b)phi}e^{i(a-b)theta} tilde A_n(a) tilde A_n(b) = p(2theta). tag{7} $$ Now take the Fourier transform $(2pi)^{-1}int_0^{2pi} dphi cdots$ of both sides of (7) with respect to $phi$ to get $$ lim_{Ntoinfty}frac{1}{N}sum_{n,a} e^{i2atheta} big|tilde A_n(a)big|^2 = p(2theta) tag{8} $$ using $tilde A_n(-a) = tilde A_n(a)^*$. This can also be written $$ sum_a e^{i2atheta} f(a) = p(2theta) tag{9} $$ with $$ f(a) := lim_{Ntoinfty}frac{1}{N}sum_{n} big|tilde A_n(a)big|^2. tag{10} $$ The function $p(2theta)$ defined in (1) can be written $$ p(2theta)=frac{1}{4}+frac{1}{8}big(e^{i2theta}+e^{-i2theta}big) tag{11} $$ so the condition (9) implies that $f(a)$ is non-zero only for $ain{-1,0,1}$. Since $f(a)$ is the average over $n$ of a non-negative quantity $big|tilde A_n(a)big|^2$, this implies that $tilde A_n(a)$ can only be non-zero for $ain{-1,0,1}$ for every $n$. However, the original function $A_n(alpha)$ is equal to either $0$ or $1$ for every $alpha$, so either it is independent of $alpha$ (which obviously contradicts the requirement (2b)) or it has a discontinuous jump at one or more values of $alpha$, which contradicts the preceding statement that its Fourier transform has only a finite number of non-zero values. This completes the proof that the requirements (2a) and (2b) cannot be satisfied, so the project is impossible.

Conclusion

This demonstrates that quantum theory's prediction cannot be reproduced by any "local hidden variables" model. The name "hidden variables" refers to the functions $A_n(theta)$ and $B_n(theta)$, and the word "local" here refers to the fact that computer $A$ does not know what angles we're giving to computer $B$, and conversely.$^dagger$

$^dagger$ This is what "local" typically means in the context of hidden-variables models. The same word has other meanings in other contexts.

---

References:

[1] http://www.mrao.cam.ac.uk/~steve/maxent2009/images/bell.pdf

[2] Section 3.1 in "Bell's theorem: experimental tests and implications" (http://physics.oregonstate.edu/~ostroveo/COURSES/ph651/Supplements_Phys651/RPP1978_Bell.pdf)