Bohr set ($B(\Gamma ,\rho )$)

For a finite abelian group $G$, frequencies $\Gamma \subseteq \hat{G}$, and $\rho >0$, the Bohr set isIts rank is $|\Gamma |$ and its width is $\rho$.

For a cyclic group $G$, a rank-$d$ phase-radius-$\rho$ Bohr set contains a symmetric proper generalized arithmetic progression of rank at most $d+1$ and cardinality at least $c(d,\rho )|G|$. The constant can be taken to depend only on $d$ and $\rho$. This geometry-of-numbers consequence follows by applying successive minima to the lattice recording the simultaneous character congruences; any cyclic subgroup factor can be represented by one additional progression coordinate. It is stronger than finding just one long ordinary progression.

A Bohr set defined by distance of each character phase to the nearest integer. For $0\le ϵ\le 1/2$, it corresponds to the chord-distance radius $2\sin (\pi ϵ)$. Numerical radius constants depend on the convention. In a group of prime order and with $d=|R|\ge 1$, simultaneous torus approximation yields a nonzero element when $ϵ>N^{-1/d}$ and an arithmetic progression of length at least $\min (N,\lceil ϵN^{1/d}\rceil )$.

If $B=B(\Gamma ,\rho )$ is a Bohr set, its $\delta$-dilate is $B_{\delta }=B(\Gamma ,\delta \rho )$. The triangle inequality on the unit circle gives $B_{\delta }+B_{\eta }\subseteq B_{\delta +\eta }$.

A rank-$d$ Bohr set $B$ is regular when small changes of its width produce proportionally small changes of its size: for $|\kappa |\le c/d$,Every Bohr set has a regular dilate $B_{\lambda }$ with $1/2\le \lambda \le 1$.

For a Bohr set in chord-distance convention, with rank $d$ and width $0<\rho \le 2$,Partition each character circle into $M=\lceil 2\pi /\rho \rceil$ arcs. One of the $M^d$ resulting boxes contains at least $|G|/M^d$ images. Subtract a fixed group element in that box from all the others. Their character differences have chord distance at most $\rho$, giving that many distinct points in the Bohr set. Since $M\le 8/\rho$ in this width range, the displayed estimate follows.

For $\rho >2$ every character constraint is automatic and the Bohr set is all of $G$. One may therefore replace $\rho$ by $\min (\rho ,2)$ in a bound valid at every width. Without a width restriction the original right side could exceed $|G|$.

Let $G=\mathbb{Z}/N\mathbb{Z}$ have prime order, and let $B(\Gamma ,\rho )$ use chord width $0<\rho \le 2$ and positive rank $d=|\Gamma |\ge 1$. Simultaneous character approximation gives a centered arithmetic progression inside it of length at leastThe prime-order condition makes every nonzero step have order $N$, so the required short arithmetic progression can use distinct residues. If $\Gamma$ is empty, the Bohr set is the whole group and contains an arithmetic progression of $N$ distinct residues; the expression $N^{1/d}$ should not be used with $d=0$. For larger chord widths the Bohr set is again the whole group. The nonwrapping progression in a cyclic Bohr set supplies a weaker exponent that also works at composite orders.

In a cyclic group of arbitrary order $q$, a Bohr set with $d$ frequencies, defined by $\Vert rx/q\Vert \le 1/16$, contains an arithmetic progression of distinct residues of length at least $q^{1/(d+1)}/32$. Partition the $d$-dimensional cube into $Q^d$ boxes, with $Q=\lfloor q^{1/(d+1)}\rfloor$. The pigeonhole principle supplies $1\le t\le Q^d$ with $\Vert rt/q\Vert \le 1/Q$ simultaneously. The residues $jt$, $0\le j\le \lfloor Q/16\rfloor$, remain in the Bohr set and do not wrap around the cyclic group. The weaker exponent permits composite moduli without assuming that every nonzero step has large order.