Transference principle in additive combinatorics

A transference principle replaces a function bounded by a sparse pseudorandom majorant with a bounded dense model that is indistinguishable by a chosen family of tests. Results for bounded functions can then be transferred to the sparse setting.

For a family $\mathcal{F}$ of real-valued functions on a finite set and normalized inner product $⟨\cdot ,\cdot ⟩$, the test-function seminorm isIt measures the largest correlation of $g$ with an allowed test. It is a seminorm, and becomes a norm when the tests separate points.

The dual test-function norm isIf $\mathcal{F}$ is closed, convex, and symmetric, the Bipolar theorem for a dual pair identifies its dual unit ball with $\mathcal{F}$. Submultiplicativity under pointwise products allows a polynomial in one test function to remain controlled in this norm.

Let $\mathcal{F}$ be a closed, convex, symmetric family of functions into $[-1,1]$ that contains the constant function $1$, and suppose its dual test-function norm is submultiplicative under pointwise products. If a nonnegative majorant $\nu$ has average at most one and $\Vert \nu -1{\Vert }_{\mathcal{F}}$ is exponentially small in $1/\epsilon$, then every $0\le g\le \nu$ has a dense model $0\le \tilde{g}\le 1$ satisfyingThe proof separates $g$ from the convex set of dense models, then approximates the positive part of the separating test by a polynomial. Submultiplicativity controls every power in that polynomial.

On any fixed compact interval, the positive part of a real-valued function $t_{+}=\max (0,t)$ can be approximated uniformly by a real polynomial. Quantitative dense-model arguments use a version whose degree and coefficient growth are explicitly controlled in terms of the approximation error.