Research proof index

Among the first $n+1$ enumerated phases, enumΘ'ArgMax returns the smallest index $i$ at which the function $g$ attains its maximum at $x$. Equivalently, every index $j\le n$ has value at most the value at $i$, and every earlier index $j<i$ has strictly smaller value.

Let $1 < q \le 2$ and let $q'$ be its Hölder conjugate. In the project’s cancellative metric-space setting, assume the associated nontangential operators satisfy the required uniform $L^2$ bound. If $F$ and $G$ are measurable and $f$ is measurable with $\lVert f(x)\rVert \le \mathbf{1}_F(x)$, then the Carleson operator obeys the restricted estimate

$\int_G^+ \mathcal{C}_K f(x)\,dx \le C(a,q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$

The polynomial Freiman-Ruzsa (PFR) conjecture: if A is a subset of an elementary abelian 2-group of doubling constant at most K, then A can be covered by at most 2 * K ^ 12 cosets of a subgroup of cardinality at most |A|.

Polynomial Freiman-Ruzsa theorem without a finite ambient-group assumption. Let $A$ be a nonempty finite subset of an elementary abelian $2$-group. If $|A+A|\le K|A|$, then there are a finite subspace $H$ and a finite set $c$ such that $A\subseteq c+H$, $|c|<2K^{12}$, and $|H|\le|A|$.

A uniform distribution on a set with doubling constant K has self Rusza distance at most log K.