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Additive combinatorics studies how the sizes and representation functions of sumsets reveal algebraic structure in subsets of abelian groups.

Apply the preceding Fourier argument to $f=1_A$, retaining the frequencies needed to make the oscillation of $1_A\ast 1_A\ast 1_A$ strictly smaller than its mean ${\alpha }^3$. The standard optimized cutoff gives a set $\Gamma \subseteq \hat{G}$ withsuch that the nonnegative function $F=1_A\ast 1_A\ast 1_A$ cannot fall from a maximal value to zero under any shift in $B(\Gamma ,\alpha )$. If $x$ maximizes $F$, thenThe support of a convolution of indicator functions is the corresponding sumset, so

Solved by gpt-5.6-sol high.