We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the $L^q$ dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of $\times p$- and $\times q$-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an $L^q$ density for all finite $q$, outside of a zero-dimensional set of exceptions.

The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to $L^q$ norms, and likewise relies on an inverse theorem for the decay of $L^q$ norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and it is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemerédi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.

For arithmetic applications, we extend and refine our previously published results to allow ramifications in a minimal way. Starting with a possibly ramified quadratic extension $F'/F$ of function fields over a finite field in odd characteristic, and a finite set of places $\Sigma$ of $F$ that are unramified in $F'$, we define a collection of Heegner--Drinfeld cycles on the moduli stack of $\mathrm{PGL}_2$-Shtukas with $r$-modifications and Iwahori level structures at places of $\Sigma$. For a cuspidal automorphic representation $\pi$ of $\mathrm{PGL}_2(\mathbb{A}_F)$ with square-free level $\Sigma$, and $r\in \mathbb{Z}_{\ge 0}$ whose parity matches the root number of $\pi_{F'}$, we prove a series of identities between

(1) the product of the central derivatives of the normalized $L$-functions $$\mathscr{L}^{(a)}\left(\pi, \frac{1}{2}\right)\mathscr{L}^{(r-a)}\left(\pi\otimes\eta, \frac{1}{2}\right), $$where $\eta$ is the quadratic id\'{e}le class character attached to $F'/F$, and $0\le a\le r$;

(2) the self intersection number of a linear combination of Heegner--Drinfeld cycles.

In particular, we can now obtain global $L$-functions with odd vanishing orders. These identities are function-field analogues of the formulae of Waldspurger and Gross--Zagier for higher derivatives of $L$-functions.

On an abelian scheme over a smooth curve over $\overline{\mathbb{Q}}$ a symmetric relatively ample line bundle defines a fiberwise Néron--Tate height. If the base curve is inside a projective space, we also have a height on its $\overline{\mathbb{Q}}$-points that serves as a measure of each fiber, an abelian variety. Silverman proved an asymptotic equality between these two heights on a curve in the abelian scheme. In this paper we prove an inequality between these heights on a subvariety of any dimension of the abelian scheme. As an application we prove the Geometric Bogomolov Conjecture for the function field of a curve defined over $\overline{\mathbb{Q}}$. Using Moriwaki's height we sketch how to extend our result when the base field of the curve has characteristic $0$.

In this paper we show that every set $A\subset \mathbb{N}$ with positive density contains $B+C$ for some pair $B,C$ of infinite subsets of $\mathbb{N}$, settling a conjecture of Erdős. The proof features two different decompositions of an arbitrary bounded sequence into a structured component and a pseudo-random component. Our methods are quite general, allowing us to prove a version of this conjecture for countable amenable groups.