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A nonempty topological space is irreducible when it is not the union of two proper closed subsets. Equivalently, every two nonempty open subsets intersect.

An algebraic variety is a geometric space locally described by polynomial equations, together with its regular functions.

A Cartier divisor is locally represented by nonzero rational functions whose ratios on overlaps are regular units. It is principal when one global rational function represents all local data.

A Weil divisor on a Noetherian integral scheme is a finite integer combination of integral codimension-one closed subschemes.

A ringed space $(X,{\mathcal{O}}_X)$ is a topological space $X$ equipped with a sheaf of rings ${\mathcal{O}}_X$. A morphism consists of a continuous map $f:X\to Y$ and a compatible morphism $f^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$.

An associative algebra is an algebra over a field whose multiplication satisfies $(ab)c=a(bc)$. Unless stated otherwise, the algebras considered here have a multiplicative identity.

A Hopf algebra is a compatible algebra and coalgebra equipped with a counit and antipode. Coordinate rings of affine algebraic groups are commutative Hopf algebras.

For a vector space with quadratic form $Q$, its Clifford algebra is generated by vectors subject to $v^2=Q(v)1$. In a basis this is equivalently ${\gamma }^{\mu }{\gamma }^{\nu }+{\gamma }^{\nu }{\gamma }^{\mu }=2{\eta }^{\mu \nu }1$.

A $k$-algebra is finitely generated when finitely many elements generate it under addition, multiplication, and scalar multiplication. Equivalently, it is a quotient of a polynomial ring $k[x_1,\dots ,x_n]$.

For birth rate $b(a)$, death rate $d(a)$ and exponential population growth rate $r$, the Euler-Lotka equation is

The Hubble time $H_0^{-1}$ is the expansion timescale inferred from the present Hubble parameter.

A Freiman homomorphism preserves additive quadruples: whenever $x_1+x_2=x_3+x_4$ in its domain, its values satisfy $\varphi (x_1)+\varphi (x_2)=\varphi (x_3)+\varphi (x_4)$.

The Gowers uniformity norm measures the average multiplicative derivative of a function around affine cubes. The $U^k$ norm detects polynomial phases of degree below $k$.

An almost period of a function $F$ in a normed space is a shift $t$ for which $\Vert {\tau }_{t}F-F\Vert$ is small.

For a finite abelian group $G$, frequencies $\Gamma \subseteq \hat{G}$, and $\rho >0$, the Bohr set isIts rank is $|\Gamma |$ and its width is $\rho$.

If $A$ has positive density in a finite abelian group, then $A+A-A-A$ contains a structured neighbourhood of zero. In a finite-dimensional vector space this neighbourhood can be taken to be a large vector subspace; in a cyclic group it can be taken to be a Bohr set.

The additive energy of a finite set $S\subseteq G$ in an abelian group $G$ counts additive quadruples, with order and repeated members allowed:At a fixed sum $s$, independently choosing two of its $r_{S+S}(s)$ ordered pairs gives $r_{S+S}(s{)}^2$ additive quadruples. Thus additive energy weights heavily represented sums more strongly than the cardinality of the sumset does.

For $A=\{0,1,2,3\}$ and $B=\{0,1,4,10\}$, the sumset representation functions giveThe arithmetic progression has additional coincidences such as $0+3=1+2$. The set $B$ still has coincidences forced by commutativity, including $0+1=1+0$, so its additive energy is not merely the sixteen identical-pair comparisons.

The same additive energy can be counted using the difference set. Define $r_{S-S}(t)=|\{(a,b)\in S\times S:a-b=t\}|$. The identity $a+b=c+d$ is equivalent to $a-c=d-b$, and the coordinate rearrangement $(a,b,c,d)\mapsto (a,c,d,b)$ is a bijection. ConsequentlyFor $A$, the difference counts from $-3$ through $3$ are again $1,2,3,4,3,2,1$. For $B$, the zero difference has four representations and each of the twelve nonzero differences has one, giving $4^2+12\cdot 1^2=28$. The sumset representation functions for sums and differences can look different while their squared totals agree.

For every nonempty finite set $S\subseteq G$ in an abelian group $G$,The lower bound follows from the Cauchy-Schwarz inequality and the total sumset representation function count:For the upper bound, any three members $a,b,c$ determine at most one fourth member $d=a+b-c$ of an additive quadruple. See Yufei Zhao's MIT lecture 25, pages 1-3, for representation counts and the energy bounds.

In the two integer examples, the lower bounds are $256/7$ and $256/10$, and the upper bound is $64$. For the subgroup $H=\{0,2,4\}$ of $\mathbb{Z}/6\mathbb{Z}$, each element of $H$ has three sum representations, so $E(H)=3\cdot 3^2=27$ and both bounds are equalities.

For nonempty $S$, dividing additive energy by $|S{|}^3$ gives a relative measure between $1/K(S)$ and one. Small doubling constant therefore forces relatively large additive energy. A converse requires care: the Balog-Szemerédi-Gowers theorem extracts a large subset with small doubling constant, rather than asserting that all of $S$ has small doubling constant.

The doubling constant of a nonempty finite set $S\subseteq G$ in an abelian group $G$ isIt measures the cardinality of the sumset relative to the original set. For the integer examples, $K(A)=7/4$ and $K(B)=10/4=5/2$. A nonconstant arithmetic progression with $n$ members has doubling constant $(2n-1)/n$, which stays below two as $n$ grows. This is the sense in which its sumset has small growth.

In any abelian group, $K(S)\ge 1$: fixing $s_0\in S$ puts a translated copy $s_0+S$ inside $S+S$. The stronger integer bound gives $K(S)\ge 2-1/|S|$ only in that setting. A finite subgroup $H$ has $K(H)=1$, as does every coset $x+H$, since its sumset is $2x+H$. In particular $H=\{0,2,4\}\subseteq \mathbb{Z}/6\mathbb{Z}$ has doubling constant one.

Small doubling constant is useful because it constrains repeated growth, as in the Plünnecke-Ruzsa inequality. It signals structure such as arithmetic progressions and subgroups; precise descriptions depend on the ambient abelian group and are the subject of the Freiman-Ruzsa theorem. The additive energy bound below gives a simpler immediate consequence: $E(S)\ge |S{|}^3/K(S)$.

For finite sets $A,B\subseteq G$ in an abelian group $G$, their difference set isHere $-b$ is the additive inverse in $G$. The difference set records possible displacements from a member of $B$ to a member of $A$; it is distinct from the set difference $A\setminus B$.

For the integer sets $A=\{0,1,2,3\}$ and $B=\{0,1,4,10\}$,The positive differences for $B$ are $1,3,4,6,9,10$, each obtained once; reversing each ordered pair supplies the corresponding negative difference. Every nonempty finite set $S$ has $0\in S-S$, with $|S|$ representations $(s,s)$. Swapping the members of an ordered pair also shows $S-S=-(S-S)$. In the cyclic group $\mathbb{Z}/6\mathbb{Z}$, the subgroup $H=\{0,2,4\}$ satisfies $H-H=H$ as well as $H+H=H$.

For finite sets $A,B\subseteq G$ in an abelian group $G$, their sumset isEach distinct value occurs only once in this set, even if many ordered pairs produce it. When $B=A$, repetitions are allowed: $a+a$ belongs to $A+A$. For the two integer sets introduced above,For $A$, the value $3$ comes from $(0,3),(1,2),(2,1),(3,0)$. For $B$, two ordered pairs give the same sum only when one reverses the other, apart from a pair being identical to itself. Thus its four pairs $(b,b)$ and six pairs with distinct members, counted without regard to order, produce ten distinct sums.

Every nonempty finite set of integers $S\subseteq \mathbb{Z}$ satisfies $|S+S|\ge 2|S|-1$. To see this, list its members as $s_1<\cdots <s_n$. The $2n-1$ distinct sumsall belong to $S+S$. A nonconstant arithmetic progression attains this bound. The ordered list is essential: in the cyclic group $\mathbb{Z}/6\mathbb{Z}$, the subgroup $H=\{0,2,4\}$ instead satisfies $H+H=H$, of cardinality three. Addition wraps around, for example $4+4=2$ in this group.