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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.

A finite sum is the result of repeated addition of its terms, written for example $a_1+\cdots +a_n={\sum }_{j=1}^n{a}_j$. The empty sum is zero. In an abelian group, associativity and commutativity make a finite sum independent of its grouping and the order of its terms.

For an infinite sequence of real or complex terms, a sum requires a limit of a sequence: the partial sums $S_n={\sum }_{j=1}^n{a}_j$ must converge to a finite value. This is a convergent series. For example, ${\sum }_{j=1}^{\infty }{2}^{-j}=1$, while ${\sum }_{j=1}^{\infty }1$ has no finite sum. Writing a summation symbol alone does not establish convergence.

An additive inverse of $x$ is an element $-x$ whose sum with $x$ is the additive identity: $x+(-x)=(-x)+x=0$. Every element of an additive group has an additive inverse.

The associative property proves uniqueness. If $y+x=0$ and $x+z=0$, thenFor example, the additive inverse of the integer $3$ is $-3$. The need for an inverse depends on the ambient system: $3$ has no additive inverse among the natural numbers, so addition on the natural numbers does not form a group.

A sphere of density ${\rho }_s$ falling in inviscid fluid of density $\rho$ has effective inertia $({\rho }_s+\rho /2)V$ and net weight $({\rho }_s-\rho )Vg$. Hence

The action $(2\pi {)}^{-1}\oint pdq$ is the area enclosed by a periodic phase-plane orbit divided by $2\pi$.

For a finite duct segment of phase length $\lambda$ joined to a semi-infinite medium, continuity of pressure and velocity gives the outgoing potential amplitudeAt half-integer $\lambda /\pi$, a large impedance mismatch can strongly enhance transmitted flux for a prescribed piston displacement.

A Beltrami flow has vorticity parallel to its velocity, expressed as $\mathbf{\omega }=\nabla \times \mathbf{u}=\lambda \mathbf{u}$. The proportionality factor can depend on position. For an incompressible flow, taking the divergence gives $\mathbf{u}\cdot \nabla \lambda =0$, so the factor is constant along each streamline.

A concrete example is $\mathbf{u}=U(\sin kz,\cos kz,0)$ for constants $U$ and $k$. Direct differentiation gives $\nabla \cdot \mathbf{u}=0$ and $\nabla \times \mathbf{u}=k\mathbf{u}$. The velocity direction varies between horizontal planes, while individual particles remain in their initial plane.

The alignment simplifies the Euler equations for an inviscid fluid because $\mathbf{u}\times \mathbf{\omega }=0$. For a steady, constant-density flow with no body force, the vector form of those equations gives $\nabla (p/\rho +|\mathbf{u}{|}^2/2)=0$. The Bernoulli equation quantity is then spatially uniform within a connected region, a stronger property than merely being constant along separate streamlines.

A spurion is a nondynamical background field assigned transformation properties so that couplings or symmetry-breaking parameters can be treated as if they arose from symmetry-covariant fields.

Supergravity is a theory in which supersymmetry is a local symmetry, with transformation parameters depending on spacetime position. Making supersymmetry local introduces a spin-$3/2$ gauge field, the gravitino, alongside the graviton describing gravity.

The reason gravity appears is structural: two supersymmetry transformations include a translation, and local translations require a dynamical spacetime geometry. The number of supercharges and the spacetime dimension determine the allowed multiplets and interactions. Eleven-dimensional supergravity and type IIA supergravity are examples with different field content and dimensional settings.

Supergravity often serves as a low-energy effective field theory of a superstring or M-theory description. That role does not make a generic finite-derivative supergravity action a complete quantum theory at arbitrarily high energy. Its validity depends on the scales at which neglected higher-derivative and additional-state effects become important.

Soft supersymmetry breaking adds a restricted set of dimensionful interactions to an otherwise supersymmetric action. Examples include gaugino masses and scalar field squared masses. Their ultraviolet behavior preserves the absence of the new quadratic divergences that generic explicit supersymmetry breaking can introduce.

Soft terms split partner masses, making the supersymmetric mass degeneracy inapplicable to the broken spectrum. They can still cause finite and logarithmic sensitivity to large partner masses, so softness does not make every choice natural.^[ref]

Four-dimensional $\mathcal{N}=1$ supersymmetry has one Weyl-spinor supercharge and four real supercharges.

Gauge-gravity duality relates a quantum field theory description to a description involving dynamical gravity, usually with additional string degrees of freedom. In the best-understood AdS/CFT correspondence, the field theory lives on the spacetime boundary and the gravitational description has an extra radial direction, possibly together with compact internal dimensions.

The two descriptions assign different variables to the same observables: a field-theory local field operator corresponds to a bulk field, and a boundary correlation function can be computed from a bulk response. A weakly curved classical bulk may describe strongly coupled field-theory dynamics when the required large-rank and coupling limits hold. The utility is therefore calculational as well as conceptual, but the existence of a field theory does not automatically guarantee a simple classical gravitational dual.

Bosonic string theory quantizes a relativistic string with bosonic worldsheet matter fields. In the ordinary flat critical theory, the worldsheet Weyl anomaly cancels in 26-dimensional target spacetime: the embedding fields contribute central charge $26$ and the reparameterization ghosts contribute $-26$.

Its string oscillators generate a tower of particle states. The closed-string massless sector includes a graviton, a Kalb–Ramond field and a dilaton, making the theory a useful model of how gravity emerges from a string spectrum. Its ground state is a tachyon, indicating an instability of the chosen vacuum rather than ordinary faster-than-light signalling. The absence of spacetime fermions and this instability limit its direct phenomenological use, but its quantization exposes many mechanisms shared by more elaborate string theories.^[ref]

A superfluid exhibits superfluidity, with phase stiffness and a phase-gradient current. In an appropriate ordered or quasi-ordered regime it supports low-speed flow without viscous dissipation, and its long-wavelength Goldstone boson has a sound-like dispersion. True thermodynamic phase selection and Bose-Einstein condensation need separate infrared conditions.

Quantum magnetism studies collective magnetic behavior in which quantum spin states, noncommuting observables and quantum fluctuations matter. A lattice of local spins can be modeled by a Heisenberg model, whose exchange interaction favors particular relations between neighbours. Its states need not resemble a set of classical arrows pointing in definite directions.

A two-spin spin singlet state illustrates the difference: it has zero total spin while retaining nontrivial spin correlations, rather than two independently known opposite arrows. Large ordered magnets can support spin waves, whereas strong quantum fluctuations can invalidate an expansion around a chosen ordered state. Measurements of magnetic susceptibility, heat capacity and excitation spectra probe different aspects of the same interacting system. The subject is therefore about both collective order and correlated states that resist a simple ordered reference picture.