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For nonempty finite set $I$ and $A\subseteq I$, let $\alpha =|A|/|I|$ be the density of a finite subset. The balanced indicator function $f=1_A-\alpha 1_I$, extended by zero to an ambient finite group, has sum zero and vanishing zero Fourier coefficient. The sum of $f$ on a subset $P\subseteq I$ is $|A\cap P|-\alpha |P|$, measuring the excess density of a finite subset there.

This is the part-preserving homomorphism density of a four-cycle in a bipartite graph with independently uniform labels $x,x^{\prime }$ in the first part and $y,y^{\prime }$ in the second. Repeated labels are included. It equals the fourth power of the four-cycle norm of the adjacency indicator function. It differs at finite sizes from a count requiring four distinct vertices.

For every $\epsilon >0$ there is a positive integer $K(\epsilon )$ such that every finite bipartite graph has an $\epsilon$-regular pair of partitions with at most $K(\epsilon )$ cells on each side.

To prove this, give $X\times Y$ the uniform probability measure, let $f$ be the indicator function of the edge set, and for partitions $\mathcal{P},\mathcal{Q}$ define their energy byThis conditional expectation is an orthogonal projection, so $0\le \mathcal{E}\le \Vert f{\Vert }_2^2\le 1$. If the partitions are not $\epsilon$-regular, choose witnesses $A_{ij}\subseteq X_i$ and $B_{ij}\subseteq Y_j$ in every irregular cell pair and refine each $X_i$ by all its $A_{ij}$ and each $Y_j$ by all its $B_{ij}$. The refined energy exceeds the old energy byOn $A_{ij}\times B_{ij}$ the absolute value function of the mean of the difference is greater than $\epsilon$, while this rectangle occupies at least an ${\epsilon }^2$ fraction of $X_i\times Y_j$. The Cauchy-Schwarz inequality therefore contributes more than ${\epsilon }^4|X_i||Y_j|/(|X||Y|)$ from that cell pair. Irregular pairs have total weight greater than $\epsilon$, so every refinement raises the energy by more than ${\epsilon }^5$. The process stops after at most $\lceil {\epsilon }^{-5}\rceil$ refinements. If the current cell counts are $r,s$, the construction gives at most $r{2}^s,s{2}^r$ new cells, so a finite iterated bound depending only on $\epsilon$ supplies $K(\epsilon )$.

If, for every $\eta >0$, the smooth numbers up to $T$ with all prime factors at most $T^{\eta }$ have density at least $\kappa (\eta )>0$, then for fixed $\epsilon >0$ there are only $O_{\epsilon }(1)$ primes $p\le N$ with $n(p)>N^{\epsilon }$. Apply the variance form of the large sieve to the indicator function of the $N^{\epsilon }$-smooth numbers up to $N^2$. For every exceptional prime, all these numbers are nonzero quadratic residues, so half its nonzero residue classes are empty. Their variance contribution is at least one third of the square of the total count. The large sieve bounds the sum of contributions by a constant times $N^2$ times that count. The density assumption with $\eta =\epsilon /2$ finishes the argument.

The distributional derivative of this indicator function is $e_1{\mathcal{H}}^1$ on the vertical segment $x_1=1/2$, so its total variation seminorm on a domain is one. This is singular with respect to planar Lebesgue measure, whereas Sobolev space weak derivatives in $W^{1,1}$ have integrable densities. Its bounded-variation space norm is $1/2+1=3/2$.

If a strongly additive arithmetic function $g$ is supported on primes up to $y$, its $j$th uniform central moment differs from the independent Bernoulli random variables model with probabilities $1/p$ by $O(j{y}^j({\sum }_{p\le y}|g(p)|{)}^j/N)$. Expansion of divisibility indicator functions reduces the comparison to $\lfloor N/d\rfloor /N=1/d+O(1/N)$.

The languages whose length-$n$ indicator functions have Boolean circuits with at most $T(n)$ gates for sufficiently large $n$, over a fixed bounded-fan-in complete basis. The family need not be constructible by one algorithm. Polynomial-size bounds give P/poly; the truth-table upper bound for circuit size applies even to undecidable languages.

For a real-valued function $H$ on a nonempty finite Cartesian product $X\times Y$, its normalized cut norm is the maximum of $|{\mathbb{E}}_{x,y}H(x,y)1_A(x)1_B(y)|$ over $A\subseteq X,B\subseteq Y$. It measures all rectangular subset discrepancies. The convention using arbitrary test functions bounded by one is equivalent within a factor of four, by decomposing their positive and negative parts into indicator functions.

A continuous function on Cantor space is uniformly continuous. Replacing it by one constant on each length-$n$ prefix cylinder set changes it by at most its oscillation on sets of diameter $2^{-n}$. This tends uniformly to zero. Each approximant is a finite linear combination of continuous indicator functions, proving density in the supremum norm.

For a normal operator with disconnected spectrum $K=K_1\bigsqcup K_2$, the indicator function of a nonempty proper clopen component union is continuous on $K$. The continuous functional calculus makes it a nonzero proper self-adjoint linear projection commuting with $T,T^{\ast }$. Its range is therefore a nontrivial closed reducing subspace. No eigenvector is needed.

A dyadic interval is one cell in a dyadic partition. Its children are the left and right half intervals. Disjoint dyadic intervals at a fixed level produce orthogonal indicator functions and independent Brownian motion increments.

There is a constant $C<3$ such that every cap set in ${\mathbb{F}}_3^n$ has cardinality less than $C^n$.

For the polynomial method in combinatorics, useOver ${\mathbb{F}}_3$, this is the indicator function of $x+y+z=0$. On $A^3$ it is therefore a diagonal tensor with $|A|$ nonzero diagonal entries. Every monomial in its expansion has individual exponents at most two and total degree at most $2n$, so one of its three variable blocks has degree at most $2n/3$. Grouping terms according to such a block and using the slice rank of a diagonal tensor giveswhere $m_n$ is the number of $\alpha \in \{0,1,2{\}}^n$ with ${\sum }_i{\alpha }_i\le 2n/3$. If $X_i=1-{\alpha }_i$ are independent random variables uniform on $\{-1,0,1\}$, thenAny exponential upper bound for this tail probability gives $m_n\le (3-ϵ{)}^n$ and hence the result after absorbing the factor three and finitely many small dimensions into $C<3$.

A finite-measure set is a measurable set on which the ambient measure is finite. Restricting to such a set makes its indicator function belong to every Lp space with finite exponent, enabling local Radon-Nikodym theorem arguments even when the whole measure space is not sigma-finite.

Let $I=[n]$ for odd $n\ge 3$, embed it in the cyclic group $G=\mathbb{Z}/(2n+1)\mathbb{Z}$, and let $A\subseteq I$ have density of a finite subset $\alpha$. If $A$ has no nonconstant three-term arithmetic progression and $n\ge 4{\alpha }^{-2}$, its balanced indicator function $f=1_A-\alpha 1_I$ satisfiesIndeed, the normalized trilinear arithmetic progression count has the Fourier analysis on a finite abelian group formula $\Lambda (h_1,h_2,h_3)={\sum }_r{\hat{h}}_1(r){\hat{h}}_2(-2r){\hat{h}}_3(r)$. Its values on $1_A$ and $\alpha 1_I$ are $\alpha n/|G{|}^2$ and ${\alpha }^3(n^2+1)/(2|G{|}^2)$. Telescoping their difference into three terms containing $f$, the Parseval identity and Cauchy-Schwarz inequality bound its magnitude by $3\max |\hat{f}|\alpha n/|G|$. The difference is at least ${\alpha }^3{n}^2/(4|G{|}^2)$, giving the claim. The zero Fourier coefficient vanishes by the definition of $\alpha$.

Normalize the conjugated cube average by the number of integer cubes whose vertices all lie in the interval. Equivalently, extend $f$ by zero to a cyclic group of order greater than $8N$ and divide its Gowers U3 norm by that of the interval's indicator function. A quadratic phase then has norm one. Omitting the denominator gives a different but uniformly equivalent interval normalization.

At level $J$, a Haar scaling function is the normalized indicator function $2^{J/2}{1}_{I_{J,k}}$ of a dyadic interval. The functions at that level form an orthonormal basis of the cellwise-constant subspace. Their normalization is what turns white-noise coefficient variances into $1/n$ and cellwise variances into $2^J/n$.

An invariant mean on a group $G$ is a positive linear functional $m:{\ell }^{\infty }(G)\to \mathbb{R}$ with $m(1)=1$ and $m(gf)=m(f)$ for every $g\in G$. Applied to indicator functions, it is a left-invariant finitely additive probability measure on all subsets of $G$.

The maximal estimate used in the classical Kakeya set problem asks, for every $\epsilon >0$, for a bound from $L^n({\mathbb{R}}^n)$ to $L^n(S^{n-1})$ with loss at most ${\delta }^{-\epsilon }$. Constants may depend on dimension and $\epsilon$, but not tube thickness or the function. Applied to an indicator function of a neighborhood of a Kakeya set, it gives the Kakeya Minkowski dimension conjecture.

For scalar functions on a finite group, normalized convolution on a finite group is $(f\ast g)(x)=|G{|}^{-1}{\sum }_{y}f(y)g(y^{-1}x)$. It is associative and need not commute. Its identity is $|G|1_{\{e\}}$, rather than the unscaled indicator function of the identity element. The Fourier analysis on a finite group convention $\hat{f}(\rho )=\mathbb{E}f(x)\rho (x)$ turns it into matrix multiplication in the same order.

The ordinary topology on $[\mathbb{N}{]}^{\omega }$ has basic sets $[s]=\{X:s⊏X\}$ for finite $s$. It is the product topology on increasing enumerations, and also the subspace topology on infinite-subset indicator functions in $\{0,1{\}}^{\mathbb{N}}$.