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The XOR lemma states that if you have a distribution $D$ on $\{0,1\}^n$, and all the Fourier coefficients of $2^n D$ are small, then it is close in $L_1$ to the uniform distribution. Specifically, suppose all Fourier coefficients are at most $\epsilon$.

The argument is a simple application of Cauchy—Schwarz with the fact Fourier preserves the $L_2$ norm to obtain:

$$|U-D|_1 \leq 2^{n/2} \epsilon$$

How tight is this? I obviously care about the exponent of $2$, not a constant outside.

I was a certain a 'probabilistic' $D$ would show tightness but that doesn't seem to be the case. Below is an explanation:

Consider generating $D$ as follows: let $D(x)$ be $\frac{1}{2^n} + z(x)\frac{\epsilon}{2^{n/2}}$ where $z(x)$ is iid on $\{-1,1\}$.

Even if we ignore this is probably not a distribution (I'm trying to point here that a probabilistic construction seems to fail so that's okay.) , this can't be turned into a tight example since there are so many Fourier coefficients, one of them will correlate significantly more in signs than the random $2^{n/2}$ with our $z$, and then we'll get more than $2^{n/2} \frac{\epsilon}{2^{n/2}}$.

Thus tightness if it exists needs to come from a controlled pseudorandom construction.

EDIT- As clement posted in a comment, bent functions are apparently a thing and clearly solve our tightness.

First notice that if $f,g$ are each bent functions on a different set of variables, then so is $fg$. Thus if there is one for $m,n$ variables, there is one for $m+n$.

Now consider $1/2-1/2x+1/2y+1/2xy$, it's a bent function in two varibles, so tensoring gives for any even $n$ gives a bent function.

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  • $\begingroup$ Sorry if this is obvious, but why are you considering $\pm\varepsilon/\sqrt{2^n}$ instead of $\pm\varepsilon/{2^n}$? (Namely, what happens if you consider a random matching of the edges of the hypercube (to sum to 1) and set a u.a.r. $z(x)=-z(y)\in\{-1,1\}$ for each (x,y)$ in the matching) $\endgroup$ – Clement C. Oct 11 at 14:48
  • $\begingroup$ As written, the function in your OP (assuming $\varepsilon \leq 1/2^{n/2}$ to have it non-negative) is not at $L_1$ distance $\varepsilon$ from uniform: each of the $2^n$ points contributes $\varepsilon/2^{n/2}$, so the $L_1$ distance is $2^{n/2}\varepsilon$. Which is consistent with the first part of your post, but is that consistent with the contradiction you claim at the end? $\endgroup$ – Clement C. Oct 11 at 14:52
  • $\begingroup$ @ClementC. You're right I should have mentioned- I'm fixing the $L1$ distance to be $2^{n/2}\epsilon$ (as mine does) and then saying if you get the bias to be $\epsilon$ then you found a tight example, but in what I showed the bias is wayy more $\endgroup$ – Andy Oct 11 at 15:31
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    $\begingroup$ Instead of a random $z$, have you considered either bent functions or $\varepsilon$-regular ones? (also, I am assuming you meant all Fourier coefficients *except the $\emptyset$ one at the beginning, since the uniform distribution itself corresponds to the constant $D$) $\endgroup$ – Clement C. Oct 11 at 19:30
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    $\begingroup$ @ClementC. Oh wow, bent functions seem like exactly what I need if I understand them correctly from the wikipedia page (the savages describe them using walsh instesad of fourier). I'm having trouble finding a simple constructions, are you aware of one? Finally, if you add this as an answer I'll accept. $\endgroup$ – Andy Oct 11 at 23:27
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Instead of choosing $z\in\{-1,1\}^{2^n}$ uniformly at random, you may want to look instead at more structured (yet "pseudo-random"-ish) functions such as bent functions:

Definition. A Boolean function $f\colon\{-1,1\}^n\to\{-1,1\}$ is called bent if $|\hat{f}(S)|=2^{-n/2}$ for all $S\subseteq [n]$.

Such functions are known to exist; see, e.g., Chapter 6.3 of Ryan O'Donnell's Analysis of Boolean functions (2014) (or the online version).

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