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Ben Inexperienced, Freddie Manners and I’ve simply uploaded to the arXiv our preprint “Sumsets and entropy revisited“. This paper makes use of entropy strategies to assault the Polynomial Freiman-Ruzsa (PFR) conjecture, which we research within the following two types:

Conjecture 1 (Weak PFR over )Let be a finite non-empty set whose doubling fixed is at most . Then there’s a subset of of density that has affine dimension (i.e., it’s contained in an affine house of dimension ).

Conjecture 2 (PFR over )Let be a non-empty set whose doubling fixed is at most . Then might be lined by cosets of a subspace of cardinality at most .

Our important outcomes are then as follows.

Theorem 3If with , then

The skew-dimension of a set is a amount smaller than the affine dimension which is outlined recursively; the exact definition is given within the paper, however suffice to say that singleton units have dimension , and a set whose projection to has skew-dimension at most , and whose fibers in have skew-dimension at most for any , can have skew-dimension at most . (The truth is, the skew-dimension is principally the most important amount which obeys all of those properties.)

Half (i) of this theorem was implicitly confirmed by Pálvölgi and Zhelezov by a special methodology. Half (ii) with changed by was established by Manners. To our data, half (iii) is totally new.

Our proof technique is to ascertain these combinatorial additive combinatorics outcomes by utilizing entropic additive combinatorics, by which we change units with random variables , and cardinality with (the exponential of) Shannon entropy. That is with the intention to make the most of some superior options of entropic additive combinatorics, most notably good conduct with respect to homomorphisms.

As an illustration, the analogue of the combinatorial doubling fixed of a finite non-empty subset of an abelian group , is the entropy doubling fixed

of a finitely-valued random variable in , the place are impartial copies of and denotes Shannon entropy. There may be additionally an analogue of the Ruzsa distance

between two finite non-empty subsets of , particularly the entropic Ruzsa distance

the place are impartial copies of respectively. (Truly, one factor we present in our paper is that the independence speculation might be dropped, and this solely impacts the entropic Ruzsa distance by an element of three at worst.) Lots of the outcomes about sumsets and Ruzsa distance have entropic analogues, however the entropic variations are barely higher behaved; as an illustration, we’ve got a contraction property

at any time when is a homomorphism. The truth is we’ve got a refinement of this inequality by which the hole between these two portions can be utilized to manage the entropic distance between “fibers” of (by which one situations and to be fastened). Then again, there are direct connections between the combinatorial and entropic sumset portions. As an illustration, if is a random variable drawn uniformly from , then

Thus if has small doubling, then has small entropic doubling. Within the converse route, if has small entropic doubling, then is shut (in entropic Ruzsa distance) to a uniform random variable drawn from a set of small doubling; a model of this assertion was confirmed in an previous paper of myself, however we set up right here a quantitatively environment friendly model, established by rewriting the entropic Ruzsa distance by way of sure Kullback-Liebler divergences.

Our first important result’s a “99% inverse theorem” for entropic Ruzsa distance: if is small enough, then there exists a finite subgroup of such that

This consequence makes use of the outcomes simply talked about to narrate to a set of small doubling, which might then be associated to a subgroup by normal inverse theorems; this offers a weak model of (1) (roughly talking shedding a sq. root within the certain), and a few extra evaluation is required to bootstrap this preliminary estimate again to (1).

We now sketch how these instruments are used to show our important theorem. For (i), we cut back issues to establishing the next bilinear entropic analogue: given two non-empty finite subsets of , one can discover subsets , with

such that have skew-dimension at most , for some absolute fixed . This may be proven by an induction on (say). One applies a non-trivial coordinate projection to . If and are very shut in entropic Ruzsa distance, then the 99% inverse theorem reveals that these random variables should every focus at some extent (as a result of has no non-trivial finite subgroups), and might cross to a fiber of those factors and use the induction speculation. If as an alternative and are far aside, then by the conduct of entropy underneath projections one can present that the fibers of and underneath are nearer on common in entropic Ruzsa distance of and themselves, and one can once more proceed utilizing the induction speculation.

For components (ii) and (iii), we first use an entropic model of an statement of Manners that units of small doubling in have to be irregularly distributed modulo . A clear formulation of this in entropic language is the inequality

at any time when take values in a torsion-free abelian group similar to ; this seems to observe from two functions of the entropy submodularity inequality. One corollary of this (and the conduct of entropy underneath projections) is that

That is the important thing hyperlink between the and worlds that’s used to show (ii), (iii): whereas (iii) depends on the nonetheless unproven PFR conjecture over , (ii) makes use of the unconditional progress on PFR by Konyagin, as detailed in this survey of Sanders. The argument has the same inductive construction to that used to ascertain (i) (and if one is prepared to exchange by then the argument is in truth comparatively easy and doesn’t want any deep partial outcomes on the PFR).

As one byproduct of our evaluation we additionally receive an interesting entropic reformulation of Conjecture 2, particularly that if is an -valued random variable then there exists a subspace of such that

Proper now the very best consequence on this route is

for any , by utilizing Konyagin’s partial consequence in direction of the PFR.

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