Research proof index

A special case of Young's convolution inequality.

For a complex-valued function and a nonzero even integer $n$, discrete self-convolution has no larger $L^n$ norm than discrete self-difference-convolution: $\|f*f\|_n\le\|f\mathbin{\circleddash}f\|_n$.

A dependent-random-choice estimate. For $p \ge 2$, a nonnegative function $f$, nonempty $A$, and intersecting sets $B_1,B_2$, the support hypothesis produces subsets $A_1 \subseteq B_1$ and $A_2 \subseteq B_2$ whose normalized difference convolution has controlled correlation with $f$. Both relative sizes $|A_i|/|B_i|$ are bounded below by the same explicit quantity, namely one quarter of a normalized $2p$-th power of the weighted $L^p$ norm of $1_A \mathbin{\circleddash} 1_A$.

Let $X_1',X_2'$ be independent copies of the $\tau$-minimizers $X_1,X_2$. Write $k=d[X_1;X_2]$ and $I_1=I[X_1+X_2:X_1'+X_2\mid X_1+X_2+X_1'+X_2']$. Then the entropy of the four-variable sum obeys $H[X_1+X_2+X_1'+X_2']\le\tfrac12H[X_1]+\tfrac12H[X_2]+(2+\eta)k-I_1$.

entropic_PFR_conjecture: For two $G$-valued random variables $X^0_1, X^0_2$, there is some subgroup $H \leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \le 11 d[X^0_1;X^0_2]$.

In the project's entropic PFR package with parameter $\eta=1/9$, there is a subspace $H$ and a random variable $U$ uniformly distributed on $H$ such that each reference variable is within six times their mutual Ruzsa distance of $U$: $d(X_1,U),d(X_2,U)\le6d(X_1,X_2)$.