Let $A$ be a subset of a finite abelian group such that $A$ has a small difference set $A-A$ and the density of $A$ is small. We prove that, counter--intuitively, the smallness (in terms of $|A-A|$) of the Fourier coefficients of $A$ guarantees that $A$ is correlated with a large Bohr set. Our bounds on the size and the dimension of the resulting Bohr set are close to exact.
Let G be an arbitrary Abelian group and let A be a finite subset of G. A has small additive doubling if |A+A| < K|A| for some K>0. These sets were studied in papers of G.A. Freiman, Y. Bilu, I. Ruzsa, M.C.--Chang, B. Green and T.Tao. In the article we prove that if we have some minor restrictions on K then for any set with small doubling there exists a set Lambda, |Lambda| << K log |A| such that |A\cap Lambda| >> |A| / K^{1/2 + c}, where c > 0. In contrast to the previous results our theorem is nontrivial for large K. For example one can take K equals |A|^η, where η>0. We use an elementary met…
We prove that every set $A\subset\mathbb{Z}/p\mathbb{Z}$ with $\mathbb{E}_x\min(1_A*1_A(x),t)\le(2+δ)t\mathbb{E}_x 1_A(a)$ is very close to an arithmetic progression. Here $p$ stands for a large prime and $δ,t$ are small real numbers. This shows that the Vosper theorem is stable in the case of a single set.
We show that if a finite, large enough subset A of an arbitrary abelian group satisfies the small doubling condition |A + A| < (log |A|)^{1 - epsilon} |A|, then A must contain a three-term arithmetic progression whose terms are not all equal, and A + A must contain an arithmetic progression or a coset of a subgroup, either of which of size at least exp^[ c (log |A|)^{delta} ]. This extends analogous results obtained by Sanders and, respectively, by Croot, Laba and Sisask in the case where the group is that of the integers or a finite field.
For $δ>0$ sufficiently small and $A\subset \mathbb{Z}^k$ with $|A+A|\le (2^k+δ)|A|$, we show either $A$ is covered by $m_k(δ)$ parallel hyperplanes, or satisfies $|\widehat{\operatorname{co}}(A)\setminus A|\le c_kδ|A|$, where $\widehat{\operatorname{co}}(A)$ is the smallest convex progression (convex set intersected with a sublattice) containing $A$. This generalizes the Freiman-Bilu $2^k$ theorem, Freiman's $3|A|-4$ theorem, and recent sharp stability results of the present authors for sumsets in $\mathbb{R}^k$ conjectured by Figalli and Jerison.
We study the number of $s$-element subsets $J$ of a given abelian group $G$, such that $|J+J|\leq K|J|$. Proving a conjecture of Alon, Balogh, Morris and Samotij, and improving a result of Green and Morris, who proved the conjecture for $K$ fixed, we provide an upper bound on the number of such sets which is tight up to a factor of $2^{o(s)},$ when $G=\mathbb{Z}$ and $K=o(s/(\log n)^3)$. We also provide a generalization of this result to arbitrary abelian groups which is tight up to a factor of $2^{o(s)}$ in many cases. The main tool used in the proof is the asymmetric container lemma, introdu…
In this paper, we study the linear structure of sets $A \subset \mathbb{F}_2^n$ with doubling constant $σ(A)<2$, where $σ(A):=\frac{|A+A|}{|A|}$. In particular, we show that $A$ is contained in a small affine subspace. We also show that $A$ can be covered by at most four shifts of some subspace $V$ with $|V|\leq |A|$. Finally, we classify all binary sets with small doubling constant.
In this paper we prove that every set $A\subset\mathbb{Z}$ satisfying the inequality $\sum_{x}\min(1_A*1_A(x),t)\le(2+δ)t|A|$ for $t$ and $δ$ in suitable ranges, then $A$ must be very close to an arithmetic progression. We use this result to improve the estimates of Green and Morris for the probability that a random subset $A\subset\mathbb{N}$ satisfies $|\mathbb{N}\setminus(A+A)|\ge k$; specifically we show that $\mathbb{P}(|\mathbb{N}\setminus(A+A)|\ge k)=Θ(2^{-k/2})$.
In the paper we prove that any sumset or difference set has large E_3 energy. Also, we give a full description of families of sets having critical relations between some kind of energies such as E_k, T_k and Gowers norms. In particular, we give criteria for a set to be a 1) set of the form H+L, where H+H is small and L has "random structure", 2) set equals a disjoint union of sets H_j, each H_j has small doubling, 3) set having large subset A' with 2A' is equal to a set with small doubling and |A'+A'| \approx |A|^4 / \E(A).
We investigate the structure of finite sets $A \subseteq \Z$ where $|A+A|$ is large. We present a combinatorial construction that serves as a counterexample to natural conjectures in the pursuit of an "anti-Freiman" theory in additive combinatorics. In particular, we answer a question along these lines posed by O'Bryant. Our construction also answers several questions about the nature of finite unions of $B_2[g]$ and $B^\circ_2[g]$ sets, and enables us to construct a $Λ(4)$ set which does not contain large $B_2[g]$ or $B^\circ_2[g]$ sets.
We give a comprehensive description of the sets $A$ in finite cyclic groups such that $|2A|<\frac94|A|$; namely, we show that any set with this property is densely contained in a (one-dimensional) coset progression. This improves earlier results of Deshouillers-Freiman and Balasubramanian-Pandey.
Suppose that G is an abelian group, A is a finite subset of G with |A+A|< K|A| and eta in (0,1] is a parameter. Our main result is that there is a set L such that |A cap Span(L)| > K^{-O_eta(1)}|A| and |L| = O(K^eta log |A|). We include an application of this result to a generalisation of the Roth-Meshulam theorem due to Liu and Spencer.
The well--known Freiman--Ruzsa Theorem provides a structural description of a set $A$ of integers with $|2A|\le c|A|$ as a subset of a $d$--dimensional arithmetic progression $P$ with $|P|\le c'|A|$, where $d$ and $c'$ depend only on $c$. The estimation of the constants $d$ and $c'$ involved in the statement has been the object of intense research. Freiman conjectured in 2008 a formula for the largest volume of such a set. In this paper we prove the conjecture for a general class of sets called chains.
We prove results on the structure of a subset of the circle group having positive inner Haar measure and doubling constant close to the minimum. These results go toward a continuous analogue in the circle of Freiman's $3k-4$ theorem from the integer setting. An analogue of this theorem in $\mathbb{Z}_p$ has been pursued extensively, and we use some recent results in this direction. For instance, obtaining a continuous analogue of a result of Serra and Zémor, we prove that if a subset $A$ of the circle is not too large and has doubling constant at most $2+\varepsilon$ with $\varepsilon<10^{-4}$…
A corollary of Kneser's theorem, one sees that any finite non-empty subset $A$ of an abelian group $G = (G,+)$ with $|A + A| \leq (2-\eps) |A|$ can be covered by at most $\frac{2}{\eps}-1$ translates of a finite group $H$ of cardinality at most $(2-\eps)|A|$. Using some arguments of Hamidoune, we establish an analogue in the noncommutative setting. Namely, if $A$ is a finite non-empty subset of a nonabelian group $G = (G,\cdot)$ such that $|A \cdot A| \leq (2-\eps) |A|$, then $A$ is either contained in a right-coset of a finite group $H$ of cardinality at most $\frac{2}{\eps}|A|$, or can be co…
Generalising results of Erdős-Freud and Lindström, we prove that the largest Sidon subset of a bounded interval of integers is equidistributed in Bohr neighbourhoods. We establish this by showing that extremal Sidon sets are Fourier-pseudorandom, in that they have no large non-trivial Fourier coefficients. As a further application we deduce that, for any partition regular equation in five or more variables, every finite colouring of an extremal Sidon set has a monochromatic solution.
First, we show that the sum-free set generated by the period-doubling sequence is not $κ$-regular for any $κ\geq 2$. Next, we introduce a generalization of the period-doubling sequence, which we call the period-$k$-folding sequences. We show that the sum-free sets generated by the period-$k$-folding sequences also fail to be $κ$-regular for all $κ\geq 2$. Finally, we study the sum-free sets generated by Sturmian sequences that begin with `11', and their difference sequences.
Let A be a finite subset of a commutative additive group Z. The sumset and difference set of A are defined as the sets of pairwise sums and differences of elements of A, respectively. The well-known inequality $σ(A)^{1/2} \leq δ(A) \leq σ(A)^2,$ where $σ(A)=\frac{|A+A|}{|A|}$ is the doubling constant of A and $δ(A)=\frac{|A-A|}{|A|}$ is the difference constant of A, relates the relative sizes of the sumset and difference set of A. The exponent 2 in this inequality is known to be optimal, for the exponent 1/2 this is unknown. We determine those sets for which equality holds in the above inequal…
In this paper we determine the number and typical structure of sets of integers with bounded doubling. In particular, improving recent results of Green and Morris, and of Mazur, we show that the following holds for every fixed $λ> 2$ and every $k \geqslant (\log n)^4$: if $ω\to \infty$ as $n \to \infty$ (arbitrarily slowly), then almost all sets $A \subset [n]$ with $|A| = k$ and $|A + A| \leqslant λk$ are contained in an arithmetic progression of length $λk/2 + ω$.
For constant $r$ and arbitrary $n$, it was known that in the graph $K_r^n$ any independent set of size close to the maximum is close to some independent set of maximum size. We prove that this statement holds for arbitrary $r$ and $n$.
Let $\mathcal{F}\subset 2^{[n]}$ be a set family such that the intersection of any two members of $\mathcal{F}$ has size divisible by $\ell$. The famous Eventown theorem states that if $\ell=2$ then $|\mathcal{F}|\leq 2^{\lfloor n/2\rfloor}$, and this bound can be achieved by, e.g., an `atomic' construction, i.e. splitting the ground set into disjoint pairs and taking their arbitrary unions. Similarly, splitting the ground set into disjoint sets of size $\ell$ gives a family with pairwise intersections divisible by $\ell$ and size $2^{\lfloor n/\ell\rfloor}$. Yet, as was shown by Frankl and Od…
In this paper we derive new upper bounds for the densities of measurable sets in R^n which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions 2,..., 24. This gives new lower bounds for the measurable chromatic number in dimensions 3,..., 24. We apply it to get a new, short proof of a variant of a recent result of Bukh which in turn generalizes theorems of Furstenberg, Katznelson, Weiss and Bourgain and Falconer about set…
This paper, in french, gives a new approach to the concept of Maximally Even Sets based on discrete Fourier transform, with several elementary but interesting and previously unpublished results. Maximally Even Sets have been invented by musicologists but have been found to bear deep relationships to other areas of science, such as the Ising model in Physics. They describe economically and characterise many famous 'scales' or subsets of the cyclic group as modelised here.
