## Entropy in Data Compression, Additive Combinatorics and Probability

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## Abstract

This thesis has four parts:

In the first part, the problem of lossless data compression with side information available to both the encoder and the decoder is considered. The finite-blocklength fundamental limits of the best achievable performance are defined, in two different versions of the problem: Reference-based compression, when a single side information string is used repeatedly in compressing different source messages, and pair-based compression, where a different side information string is used for each source message. General achievability and converse theorems are established for arbitrary source-side information pairs. Nonasymptotic normal approximation expansions are proved for the optimal rate in both the reference-based and pair-based settings, for memoryless sources. These are stated in terms of explicit, finite-blocklength bounds, that are tight up to third-order terms. Extensions that go significantly beyond the class of memoryless sources are obtained. The relevant source dispersion is identified and its relationship with the conditional varentropy rate is established. Interestingly, the dispersion is different in reference-based and pair-based compression, and it is proved that the reference-based dispersion is in general smaller.

Secondly, a discrete, asymptotic analogue of the entropy power inequality due to Tao is discussed. The nonasymptotic version of the result is obtained. Applications in information theory and connections with the entropic central limit theorem are discussed.

In the third part, we recall some of the history of the information-theoretic approach to deriving core results in probability theory and give three new such proofs. A strengthened version of the central limit theorem for discrete random variables is established, relying only on information-theoretic tools and elementary arguments. It is shown that the relative entropy between the standardised sum of

In the final part, we provide a collection of inequalities for the differential entropy of an arbitrary random variable convolved with a Gaussian. In terms of techniques, as well as of the results themselves, these inequalities lie in the interface between sumset inequalities for entropy and the central limit theorem.