Some new results on the higher energies I

We prove a new class of low-energy decompositions which, amongst other consequences, imply that any finite set $A$ of integers may be written as $A = B \cup C$, where $B$ and $C$ are disjoint sets satisfying \[ |\{ (b_1, \dots, b_{2s}) \in B^{2s} \ | \ b_1 + \dots + b_{s} = b_{s+1} + \dots + b_{2s}\}| \ll_{s} |B|^{2s - (\log \log s)^{1/2 - o(1)}} \] and \[ |\{ (c_1, \dots, c_{2s}) \in C^{2s} \ | \ c_1 \dots c_{s} = c_{s+1} \dots c_{2s} \}| \ll_{s} |C|^{2s - (\log \log s)^{1/2...

We derive several new bounds for the problem of difference sets with local properties, such as establishing the super-linear threshold of the problem. For our proofs, we develop several new tools, including a variant of higher moment energies and a Ramsey-theoretic approach for the problem.

We prove that every additive set $A$ with energy $E(A)\ge |A|^3/K$ has a subset $A'\subseteq A$ of size $|A'|\ge (1-\varepsilon)K^{-1/2}|A|$ such that $|A'-A'|\le O_\varepsilon(K^{4}|A'|)$. This is, essentially, the largest structured set one can get in the Balog-Szemer\'edi-Gowers theorem.

We study higher moments of convolutions of the characteristic function of a set, which generalize a classical notion of the additive energy. Such quantities appear in many problems of additive combinatorics as well as in number theory. In our investigation we use different approaches including basic combinatorics, Fourier analysis and eigenvalues method to establish basic properties of higher energies. We provide also a sequence of applications of higher energies additive com...

We demonstrate $k+1$-term arithmetic progressions in certain subsets of the real line whose "higher-order Fourier dimension" is sufficiently close to 1. This Fourier dimension, introduced in previous work, is a higher-order (in the sense of Additive Combinatorics and uniformity norms) extension of the Fourier dimension of Geometric Measure Theory, and can be understood as asking that the uniformity norm of a measure, restricted to a given scale, decay as the scale increases. ...

In a recent breakthrough Kelley and Meka proved a quasipolynomial upper bound for the density of sets of integers without non-trivial three-term arithmetic progressions. We present a simple modification to their method that strengthens their conclusion, in particular proving that if $A\subset\{1,\ldots,N\}$ has no non-trivial three-term arithmetic progressions then \[\lvert A\rvert \leq \exp(-c(\log N)^{1/9})N\] for some $c>0$.

Two well studied Ramsey-theoretic problems consider subsets of the natural numbers which either contain no three elements in arithmetic progression, or in geometric progression. We study generalizations of this problem, by varying the kinds of progressions to be avoided and the metrics used to evaluate the density of the resulting subsets. One can view a 3-term arithmetic progression as a sequence $x, f_n(x), f_n(f_n(x))$, where $f_n(x) = x + n$, $n$ a nonzero integer. Thus a...

We show that any subset of the natural numbers with positive logarithmic Banach density contains a set that is within a factor of two of a geometric progression, improving the bound on a previous result of the authors. Density conditions on subsets of the natural numbers that imply the existence of approximate powers of arithmetic progressions are developed and explored.

The problem of constructing dense subsets S of {1,2,..,n} that contain no arithmetic triple was introduced by Erdos and Turan in 1936. They have presented a construction with |S| = \Omega(n^{\log_3 2}) elements. Their construction was improved by Salem and Spencer, and further improved by Behrend in 1946. The lower bound of Behrend is |S| = Omega({n \over {2^{2 \sqrt{2} \sqrt{\log_2 n}} \cdot \log^{1/4} n}}). Since then the problem became one of the most central, most fundame...

Given a set of integers with no three in arithmetic progression, we construct a Stanley sequence by adding integers greedily so that no arithmetic progression is formed. This paper offers two main contributions to the theory of Stanley sequences. First, we characterize well-structured Stanley sequences as solutions to constraints in modular arithmetic, defining the modular Stanley sequences. Second, we introduce the basic Stanley sequences, where elements arise as the sums of...