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2 added 2 characters in body

Some initial thoughts:

• the question is basically asking whether $f*f$ can be close to a constant multiple $c1_E$ of an indicator function (these are the only non-negative functions for which Holder is sharp).
• the hypothesis that $f$ is non-negative is going to be crucial. Note that any Schwartz function can be expressed as $f*f$ for some complex-valued f by square-rooting the Fourier transform, and so by approximating an indicator function by a Schwartz function we see that there is no gain.
• On the other hand the hypothesis that f $f$ is an indicator function (or a constant multiple thereof) is only of limited utility, because any non-negative function in $L^1$ can be expressed as the weak limit of constant multiples of indicator functions (similarly to how a grayscale image can be rendered using black and white pixels in the right proportion; the indicator is of a random union of small intervals whose intensity is proportional to $f$).
• If $f*f$ is close to $c1_E$, and we normalise $c=1$ and $|E|=1$, then $f$ has $L^1$ norm close to $1$ and the Fourier transform has $L^4$ norm close to $1$ (i.e. the Gowers $U^2$ norm is close to $1$). Using the quantitative idempotent theorem of Green and Sanders we also see that the $L^2$ norm of $f$ (which controls the Wiener norm of $f*f$) is much larger than $1$. But it's not clear to me where to go next from here.
1

Some initial thoughts:

• the question is basically asking whether $f*f$ can be close to a constant multiple $c1_E$ of an indicator function (these are the only non-negative functions for which Holder is sharp).
• the hypothesis that $f$ is non-negative is going to be crucial. Note that any Schwartz function can be expressed as $f*f$ for some complex-valued f by square-rooting the Fourier transform, and so by approximating an indicator function by a Schwartz function we see that there is no gain.
• On the other hand the hypothesis that f is an indicator function (or a constant multiple thereof) is only of limited utility, because any non-negative function in $L^1$ can be expressed as the weak limit of constant multiples of indicator functions (similarly to how a grayscale image can be rendered using black and white pixels in the right proportion; the indicator is of a random union of small intervals whose intensity is proportional to $f$).
• If $f*f$ is close to $c1_E$, and we normalise $c=1$ and $|E|=1$, then $f$ has $L^1$ norm close to $1$ and the Fourier transform has $L^4$ norm close to $1$ (i.e. the Gowers $U^2$ norm is close to $1$). Using the quantitative idempotent theorem of Green and Sanders we also see that the $L^2$ norm of $f$ (which controls the Wiener norm of $f*f$) is much larger than $1$. But it's not clear to me where to go next from here.