arXiv subjects
Publications and source records attributed to Tom Sanders.
Suppose that G is a finite group and A is a subset of G such that 1_A has algebra norm at most M. Then 1_A is a plus/minus sum of at most L cosets of subgroups of G, and L can be taken to be triply tower in O(M^2). This is a quantitative version of the non-abelian idempotent theorem.
We show that if G is a finite group and f is a {0,1}-valued function on G with Fourier algebra norm at most M then f may be computed by a coset decision tree (that is a decision tree in which at each vertex we query membership of a given coset) having at most \exp(\exp(\exp(O(M^2)))) leaves. A short calculation shows that any {0,1}-valued function which may be computed by a coset decision tree with m leaves has Fourier algebra norm at most \exp(O(m)).
We describe, with proof, an infinite class of examples of finite sets of integers $A$ for which $\|\widehat{1_A}\|_1\geq (c_\mathcal{R}-o_{|A| \rightarrow \infty}(1))\sqrt{|A|}$ where $c_\mathcal{R}=0.953\dots$.
Suppose that $G$ is a compact Hausdorff Abelian group. We say $μ\in M(G)$ is strongly continuous if $|μ|(x+H)=0$ for any $x \in G$ and any $H \leq G$ that is closed and of infinite index. We prove that for any sufficiently rapidly decreasing sequence $(a_{n})_{n=1}^{\infty}\in c_{0}(\mathbb{N})$, for every strongly continuous $μ\in M(G)$ with $\|μ\| \leq 1$ and $\widehatμ(\widehat{G})\subset \{a_n: n \in \mathbb{N}\}\cup\{0\}$, the measure $μ\astμ$ is absolutely continuous with respect to Haar measure on $G$. This implies that $μ$ does not exhibit the so-called Wiener-Pitt phenomenon. The paper is a continuation of investigations started in \cite{ow}.
We study modular analogues of Schur numbers for systems of linear equations. We show that these only depend on the number of equations, not their coefficients and in the case of one equation show stronger bounds.
We show that for any $\varepsilon>0$ if $ϕ:\mathbb{T} \rightarrow \mathbb{T}$ is continuous and $\|\exp(-2πi z ϕ)\|_{A(\mathbb{T})} =O_{|z|\rightarrow \infty}(\log^{\frac{1}{8}-\varepsilon} |z|)$ then $ϕ(x)=wx+t$ for some $w \in\mathbb{Z}$ and $t \in \mathbb{T}$.
We show that for non-zero integers $a$ and $b$ there is a natural number $N < \exp(r^{2+o_{a,b;r\rightarrow \infty}(1)})$ such that in any $r$-colouring of $\{1,\dots,N\}$ there are $x,y,z$, all in the same colour class, such that $ax-ay=bz$.
We show that if $G$ is a finite Abelian group and $f$ is an integer-valued map on $G$ with algebra norm at most $M$ then there is some $L < \exp(M^{4+o(1)})$, cosets of (possibly different) subgroups $W_1,...,W_L$, and $s_1,...,s_L \in \{-1,1\}$ such that $f=\sum_i{s_i1_{W_i}}$.
For $k \in \mathbb{N}$, write $S(k)$ for the largest natural number such that there is a $k$-colouring of $\{1,\dots,S(k)\}$ with no monochromatic solution to $x-y=z^2$. That $S(k)$ exists is a result of Bergelson, and a simple example shows that $S(k) \geq 2^{2^{k-1}}$. The purpose of this note is to show that $S(k)\leq 2^{2^{2^{O(k)}}}$.
We establish various results including the following: if $1<p<\infty$ and $σ$ is a bijection between the trigonometric functions on $[0,1)$ and the Walsh functions on $[0,1)$. Then $σ$ does not extend to an isomorphism $L_p[0,1) \rightarrow L_p[0,1)$.
We show that if $G$ is a discrete Abelian group and $A \subset G$ has $\|1_A\|_{B(G)} \leq M$ then $A$ is $O(\exp(πM))$-stable in the sense of Terry and Wolf.
Suppose that $A$ is a $k \times d$ matrix of integers and write $\mathfrak{R}_A:\mathbb{N} \rightarrow \mathbb{N}\cup \{ \infty\}$ for the function taking $r$ to the largest $N$ such that there is an $r$-colouring $\mathcal{C}$ of $[N]$ with $\bigcup_{C \in \mathcal{C}}{C^d}\cap \ker A =\emptyset$. We show that if $\mathfrak{R}_A(r)<\infty$ for all $r \in \mathbb{N}$ then $\mathfrak{R}_A(r) \leq \exp (\exp(r^{O_{A}(1)}))$ for all $r \geq 2$. When the kernel of $A$ consists only of Brauer configurations -- that is vectors of the form $(y,x,x+y,\dots,x+(d-2)y)$ -- the above has been proved by Chapman and Prendiville with good bounds on the $O_A(1)$ term.
For G a finite non-Abelian group we write c(G) for the probability that two randomly chosen elements commute and k(G) for the largest integer such that any k(G)-colouring of G is guaranteed to contain a monochromatic quadruple (x,y,xy,yx) with xy not equal to yx. We show that c(G) tends to 0 if and only if k(G) tends to infinity.
We show that there is an absolute $c>0$ such that any subset of $\mathbb{F}_2^\infty$ of size $N$ is $O(N^{1-c})$-stable in the sense of Terry and Wolf. By contrast a size $N$ arithmetic progression in the integers is not $N$-stable.
We show that if A is a finite set of integers then it has a subset S of size \log^{1+c} |A| (c>0 absolute) such that s+s' is never in A when s and s' are distinct elements of S.
We show that if $f$ is an integer-valued function with spectral norm at most $M$ then there are subspaces $V_1,\dots,V_L$ and signs $σ_1,\dots,σ_L \in \{-1,1\}$ such that $f=σ_1 1_{V_1} + \dots + σ_L 1_{V_L}$ where $L < \exp(M^{3+o(1)})$. This note extracts out the argument from arXiv:1610.07092 to the model setting of $\mathbb{F}_2^n$. The hope is that it will clarify the arguments of that paper.
We show that if f is an injection from an n-dimensional Boolean cube (considered as an additive group) to a group of prime order then the probability that f(x+y)=f(x)+f(y) is O(2^{-n/11}).
Suppose that $\mathbb{F}_p$ is coloured with $r$ colours. Then there is some colour class containing at least $c_r p^2$ quadruples of the form $(x, y , x + y, xy)$.