arXiv subjects
Publications and source records attributed to Tom Sanders.
Suppose that f is a boolean function from F_2^n to {0,1} with spectral norm (that is the sum of the absolute values of its Fourier coefficients) at most M. We show that f may be expressed as +/- 1 combination of at most 2^(2^(O(M^4))) indicator functions of subgroups of F_2^n.
We show that if A is a subset of Z/pZ (p a prime) of density bounded away from 0 and 1 then the A(Z/pZ)-norm (that is the l^1-norm of the Fourier transform) of the characterstic function of A is bounded below by an absolute constant times (log p)^{1/2 - ε} as p tends to infinity. This improves on the exponent 1/3 in recent work of Green and Konyagin.
Suppose that G is an abelian group and A is a finite subset of G containing no three-term arithmetic progressions. We show that |A+A| >> |A|(log |A|)^{1/3-ε} for all ε>0.
Suppose that A is a subset of the integers {1,...,N} of density a. We provide a new proof of a result of Green which shows that A+A contains an arithmetic progression of length exp(ca(log N)^{1/2}) for some absolute c>0. Furthermore we improve the length of progression guaranteed in higher sumsets; for example we show that A+A+A contains a progression of length roughly N^{ca} improving on the previous best of N^{ca^{2+ε}}.
We show that if A is a subset of F_2^n and |A+A| < K|A| then A is contained in a subspace of size at most 2^{O(K^{3/2}log K)}|A|. This improves on the previous best of 2^{O(K^2)}.
We show that if A is a subset of Z_4^n containing no three-term arithmetic progression in which all the elements are distinct then |A|=o(4^n/n).
Suppose that A is a subset of {1,...,N} such that the difference between any two elements of A is never one less than a prime. We show that |A| = O(N exp(-c(log N)^{1/4})) for some absolute c>0.
Suppose that A is a subset of F_2^n of density as close to 1/3 as possible. We show that the A(F_2^n)-norm (that is the sum of the absolute values of the Fourier transform) of the characterstic function of A is bounded below by an absolute constant times log n as n tends to infinity.
Suppose that G is a locally compact abelian group, and write M(G) for the algebra of bounded, regular, complex-valued measures under convolution. A measure μin M(G) is said to be idempotent if μ* μ= μ, or alternatively if the Fourier-Stieltjes transform μ^ takes only the values 0 and 1. The Cohen-Helson-Rudin idempotent theorem states that a measure μis idempotent if and only if the set {r in G^ : μ^(r) = 1} belongs to the coset ring of G^, that is to say we may write μ^ as a finite plus/minus 1 combination of characteristic functions of cosets r_j + H_j, where the H_j are open subgroups of G^. In this paper we show that the number L of such cosets can be bounded in terms of the norm ||μ||, and in fact one may take L <= \exp\exp(C||μ||^4). In particular our result is non-trivial even for finite groups.
We prove a theorem claimed in math.CA/0605519 which asserts that if A is a subset of a compact abelian group G with density of a particular (natural, although technical) form then the A(G)-norm (that is the sum of the absolute values of the Fourier transform) of the characteristic function of A cannot be too small.