Proceedings of the IISE Annual Conference & Expo 2024
A. Brown Greer, C. Contardo, J.-M. Frayret, eds. Xiaoyue Liu, Jingze Li,
Mathieu Dahan, Benoit Montreuil7118

#  Multi-Period Stochastic Logistic Hub Capacity Planning
for Relay Transportation

Xiaoyue Liu  Jingze Li  Mathieu Dahan  Benoit Montreuil
Physical Internet Center  H. Milton Stewart School of Industrial & Systems
Engineering
Georgia Institute of Technology  Atlanta  GA 30332

###### Abstract

This study focuses on relay transport carriers (RTCs) that contract with hub
providers to lease hub capacity and employ relay transportation via hubs. It
enables long-haul freight shipments to be transported by multiple short-haul
drivers commuting between fixed-base hubs, promoting a driver-friendly
approach. Inspired by Physical Internet, our paper addresses the multi-period
capacity planning of logistic hubs within relay networks, accounting for
uncertainty in demand and travel times. We model the problem as a two-stage
stochastic optimization to determine the dynamic logistic hub throughput
capacities for each planning period, ensuring the fulfillment of logistic
demand while simultaneously minimizing both hub and transportation costs. This
optimization problem falls within the NP-hard complexity class. To alleviate
the inherent challenges in solving this problem, we employ a scenario
reduction algorithm based on the fast forward selection (FFS) method to reduce
computational effort while preserving approximation quality. Experiments with
an automotive-delivery RTC in the Southeastern US demonstrate that our
capacity planning model enables RTCs to proactively respond to dynamic
circumstances, curtail avoidable expenditures, and enhance overall logistical
efficiency.

## Keywords

Multi-Period Stochastic Capacity Planning, Relay Transportation, Hub Capacity-
Routing Problem, Physical Internet

##  1 Introduction

As reported in [1], truck driver shortage reached a historic high of over
80,000 in 2021, and predictions indicate it may exceed 160,000 by 2030. The
traditional end-to-end transportation heavily depends on long-haul freight
movement. Covering extensive distances in a single trip requires drivers to
endure prolonged periods away from home, which negatively impacts the physical
and mental health of drivers, constituting a major factor in truck driver
shortage [2, 3]. To address these problems, many researchers are exploring a
shift from conventional end-to-end transportation to relay transportation [4,
5]. The key idea behind relay transportation is to position logistic hubs
along transport routes. Then the tasks of long-haul freight shipments can be
distributed among multiple short-haul drivers commuting between fixed-base
hubs. This approach enables truckers to operate near their residences and
return home everyday.

However, one of the main challenges in adopting relay transportation is the
need for a large number of physical facilities - logistic hubs, which is
especially unfriendly to small and medium-sized carriers. The introduction of
the Physical Internet (PI) offers a promising avenue for addressing this
challenge. Proposed by [4], PI presents an innovative approach to shaping
supply chains, guiding them towards sustainability, economic efficiency, and
social responsibility. One of the core concepts of PI is to promote an open
and cooperative sharing economy. Distinguishing from the ordinary sharing
economy, the PI infrastructure is open to multiple parties rather than
exclusively dedicated to a single company or group of companies. Building on
this concept, [6] utilize a multi-agent simulation to demonstrate that
resource-sharing transportation within the framework of PI enhances the
performance of transport operations across financial, operational, social, and
environmental dimensions.

Motivated by the innovative concept of PI, we examine a setting where an RTC
aims to contract with hub providers to secure space for relay transportation
under demand variations and uncertain travel time caused by network
disruptions. Within this context, we define a multi-period stochastic hub
capacity planning with routing problem (referred to as MSHCRP) to determine
dynamic hub throughput capacities as well as shipment routing at each planning
period. We then formulate a two-stage stochastic optimization model for the
MSHCRP, aiming to minimize overall expenses while addressing both demand and
travel time uncertainty. However, the resulting mixed-integer programming
(MIP) poses computational challenges for real-world problems. To tackle this
issue, we adopt a scenario reduction algorithm proposed by [7] to approximate
underlying continuous distributions for demand and travel time.

The rest of this paper is organized as follows. Section 2 briefly summarizes
literature reviews of the relevant works. In Section 3, we describe the
problem setting and state research assumptions. Section 4 formulates a two-
stage stochastic program model and presents the scenario generation and
reduction method to alleviate the computational challenge. Section 5
delineates the dataset and experiments, extracting insights from the obtained
results. Finally, Section 6 concludes this paper and discusses future research
avenues.

##  2 Related Research

The hub location-routing problem (HLRP) has received considerable attention in
academic research for decades. Introduced by [9], HLRP comprises two levels of
decisions: a strategic level focused on determining hub locations and an
operational level dealing with route planning. However, a notable critique of
classical (single-period) HLRP is the disparity between the long-term nature
of strategic decisions and the short-term nature of operational decisions
[10]. To bridge this temporal gap, the multi-period HLRP emerges as a viable
solution, offering more frequent planning intervals for routing decisions.
Although this multi-period approach is not novel, relevant research within the
HLRP literature remains limited. Previous work by [11] explored scenarios
involving progressive hub openings or closures, and studies by [12] considered
dynamic modular hub capacity that can gradually expand over the planning
horizon. These considerations reflect real-world constraints, where budgetary
limitations and resource availability may prohibit the simultaneous setup of
an entire hub network. Another issue identified in the existing literature on
HLRP is the heavy reliance on long-haul deliveries due to the significant
costs associated with constructing hubs. This results in poor working
conditions for drivers and further exacerbates the ongoing truck driver
shortage. To alleviate this problem, we suggest an innovative solution where
carriers can dynamically lease hub capacity through short-term contracts and
use these hubs as simple relay facilities, which are not only cost-effective
but also easily adjustable. This variant of multi-period HLRP allows carriers
to manage long-haul shipments efficiently using short-haul drivers, promoting
a driver-friendly approach. Additionally, it provides greater flexibility for
carriers to adapt to dynamic environments.

The increasing complexities of real-world scenarios highlight the need to
incorporate stochastic elements into problem modeling. In the literature on
HLRP, two main types of uncertainties are usually considered separately:
demand uncertainty, reflecting changes or fuzziness in demand, and travel time
uncertainty, associated with factors such as traffic congestion or network
failures. For instance, [12] utilized a polynomial function to capture time-
dependent demand dynamics, addressing a multi-period HLRP through various
meta-heuristic algorithms. [13] addressed seasonal fluctuations in the air
freight market, formulating air hub location and flight routing as a two-stage
stochastic program model. In a different context, [14] explored HLRP with
congestion effects in cargo delivery, while [15] proposed an HLRP model based
on two-stage stochastic optimization under the risk of hub disruptions.
However, to the best of our knowledge, limited effort has been made to
integrate both demand and travel time uncertainty within a multi-period
framework, let alone adapting it to the context of relay-based transportation.
Our study aims to bridge this gap in the HLRP literature by formulating a two-
stage stochastic optimization model that simultaneously addresses two distinct
uncertainties in a multi-period and relay-based setting. In the experimental
phase, we conduct numerical experiments based on real automotive data,
focusing on critical uncertainty parameters. These experiments yield valuable
insights into the impact of demand and travel time uncertainty on network
robustness, cost-effectiveness, delivery efficiency, and environmental
impacts.

##  3 Problem Description

This work tackles a logistic hub capacity planning challenge within a relay-
based transportation network under a multi-period and multi-uncertainty
environment. The key objective is to efficiently serve logistical demands
while minimizing hub costs and transportation costs under uncertain
conditions. In our study, logistic hubs primarily serve as facilities offering
parking spots for driver waiting and resting, along with switch zones and
buffer spots for driver and trailer swaps to support relay transportation.
This definition aligns with the design of PI transit centers [8]. The
contracted capacity in this scenario denotes the throughput capacity of
logistic hubs, involving modular factors such as the number of parking spots
and trailer swap docks. Specifically, we assume that the RTC makes one-year
contracts with hub providers, indicating a total planning horizon of one year.
Within this time frame, we explore a case where carriers have the flexibility
to modify their contracted throughput capacities across different planning
seasons in response to evolving demand needs, albeit with associated capacity-
changing penalties. In Section 5, we conduct experiments to delve into the
advantages of this dynamic (i.e., multi-period) setting.

Figure 1: Timeline of the multi-period stochastic capacity planning model

A primary challenge lies in accounting for different time spans for various
decision types. To address this, we formulate the problem using a two-stage
stochastic program that divides decisions into two stages: The first stage
involves logistic hub capacity level decisions, while the second stage
encompasses freight routing decisions and extra capacity decisions. As
depicted in Figure 1, capacity level decisions are made at the beginning of
each planning season $t_{k}$, whereas routing and extra capacity usage
decisions are made for each operational period $\varphi_{i}$ after realizing
demand and geographical disruptions. We assume all shipments are dispatched at
the beginning of each operational period and are fully routed through the
network, without occupying the transportation resources of the subsequent
operational period.

Another major challenge is to determine logistic hub capacity in the face of
uncertainty. When signing a capacity contract, the demand for future periods
remains unknown. However, it is the demand that primarily influences capacity
decisions. To model demand uncertainty, we assume that the carrier has
historical data reflecting demand distribution from past service records, and
model this uncertainty through probabilistic scenarios. Yet, even with
sufficient data for scenario generation, incorporating too many scenarios can
pose a computational burden. To balance this trade-off, we implement a
scenario reduction to reduce computational effort while preserving the quality
of the approximation.

##  4 Problem Formulation

We model this problem as a two-stage stochastic program to strategically
determine logistic hub throughput capacities for each planning season,
ensuring the fulfillment of logistic needs while minimizing both hub costs and
transportation costs. The first stage decisions involve determining the
throughput capacity levels of each logistic hub. In the second stage,
following the realization of demand and geographical disruptions, decisions
are made regarding freight shipment routing within the network and the
utilization of extra throughput capacity at each logistic hub. The model
notations are displayed in Table LABEL:tab:notations and its formulation are
as follows:

Table 1: Notation for the multi-period stochastic capacity planning model Sets |  | Parameters |
---|---|---|---
$\boldsymbol{O}$ |  Set of origins, indexed by $o$ | $q_{od}(\varphi,\omega)$ |  Demand from origin $o$ to destination $d$ in
$\boldsymbol{D}$ |  Set of destinations, indexed by $d$ |  |  operational period $\varphi$ under scenario $\omega$
$\boldsymbol{H}$ |  Set of logistic hubs, indexed by $h$ | $t_{a}(\varphi,\omega)$ |  Travel time on arc $a$ in operational period
$\boldsymbol{P}$ |  Set of demand OD pairs, indexed by $(o,d)$ |  |  $\varphi$ under scenario $\omega$ (affected by disruptions)
$\boldsymbol{A}$ |  Set of arcs, indexed by $a$ or $(i,j)$ | $c^{H}_{hl_{1}l_{2}}$ |  Hub cost to change capacity level from $l_{1}$ to
$\boldsymbol{T}$ |  Set of planning seasons, indexed by $t$ |  |  $l_{2}$ and operate hub $h$ at capacity level $l_{2}$
$\boldsymbol{T_{t}}$ |  Set of operational periods in season $t$, indexed by $\varphi$ | $c^{F}$ | Flow unit cost per hour
$\boldsymbol{W_{t}}$ |  Set of scenarios in season $t$, indexed by $\omega$ | $c^{P}_{h}$ |  Penalty cost for using extra capacity at hub $h$
$\boldsymbol{L}$ |  Set of capacity levels, indexed by $l$ | $u_{hl}$ |  Hub capacity at capacity level $l$ at hub $h$
$\boldsymbol{\delta^{+}(i)}$ |  Set of arcs exiting node $i$ | $m$ | Number of days during an operational period
$\boldsymbol{\delta^{-}(i)}$ |  Set of arcs entering node $i$ | $p_{\omega}$ |  Probability of scenario $\omega$
Variables |
---|---
$X_{hl_{1}l_{2}}(t)$ |  Whether to change capacity level from $l_{1}$ to $l_{2}$ at hub $h$ at the beginning of planning season $t$
$F_{oda}(\varphi,\omega)$ |  Freight volume from origin $o$ to destination $d$ on arc $a$ in period $\varphi$ under scenario $\omega$
$E_{h}(\varphi,\omega)$ |  Extra capacity usage at hub $h$ in operational period $\varphi$ under scenario $\omega$

$\displaystyle\min\sum_{t\in\boldsymbol{T}}\sum_{h\in\boldsymbol{H}}\sum_{l_{1}\in\boldsymbol{L}}\sum_{l_{2}\in\boldsymbol{L}}c_{hl_{1}l_{2}}^{H}\cdot
X_{hl_{1}l_{2}}(t)+\sum_{t\in\boldsymbol{T}}\sum_{\omega\in\boldsymbol{W_{t}}}\sum_{\varphi\in\boldsymbol{T_{t}}}p_{\omega}\left(\sum_{a\in\boldsymbol{A}}\sum_{(o,d)\in\boldsymbol{P}}c^{F}\cdot
t_{a}(\varphi,\omega)\cdot
F_{oda}(\varphi,\omega)+\sum_{h\in\boldsymbol{H}}c_{h}^{P}\cdot
E_{h}(\varphi,\omega)\right)$ (1) $\displaystyle s.t.$
$\displaystyle\sum_{a\in\boldsymbol{\delta^{+}(o)}}F_{oda}(\varphi,\omega)=q_{od}(\varphi,\omega),\quad\forall(o,d)\in\boldsymbol{P},\forall\varphi\in\boldsymbol{T}_{\boldsymbol{t}},\forall\omega\in\boldsymbol{W}_{\boldsymbol{t}},\forall
t\in\boldsymbol{T}$ (2)
$\displaystyle\sum_{a\in\boldsymbol{\delta^{-}(d)}}F_{oda}(\varphi,\omega)=q_{od}(\varphi,\omega),\quad\forall(o,d)\in\boldsymbol{P},\forall\varphi\in\boldsymbol{T}_{\boldsymbol{t}},\forall\omega\in\boldsymbol{W}_{t},\forall
t\in\boldsymbol{T}$ (3)
$\displaystyle\sum_{a\in\boldsymbol{\delta^{+}(h)}}F_{oda}(\varphi,\omega)=\sum_{a\in\boldsymbol{\delta^{-}(h)}}F_{oda}(\varphi,\omega),\quad\forall
h\in\boldsymbol{H},\forall(o,d)\in\boldsymbol{P},\forall\varphi\in\boldsymbol{T}_{\boldsymbol{t}},\forall\omega\in\boldsymbol{W}_{t},\forall
t\in\boldsymbol{T}$ (4)
$\displaystyle\sum_{(o,d)\in\boldsymbol{P}}\sum_{a\in\boldsymbol{\delta^{-}(h)}}F_{oda}(\varphi,\omega)\leq
m\cdot\left(\sum_{l_{1}\in\boldsymbol{L}}\sum_{l_{2}\in\boldsymbol{L}}u_{hl_{2}}\cdot
X_{hl_{1}l_{2}}(t)+E_{h}(\varphi,\omega)\right),\quad\forall
h\in\boldsymbol{H},\forall\varphi\in\boldsymbol{T}_{\boldsymbol{t}},\forall\omega\in\boldsymbol{W}_{\boldsymbol{t}},\forall
t\in\boldsymbol{T}$ (5)
$\displaystyle\sum_{l\in\boldsymbol{L}}X_{h0l}(0)=1,\quad\forall
h\in\boldsymbol{H}$ (6)
$\displaystyle\sum_{l_{1}\in\boldsymbol{L}}\sum_{l_{2}\in\boldsymbol{L}}X_{hl_{1}l_{2}}(t)=1,\quad\forall
h\in\boldsymbol{H},\forall t\in\boldsymbol{T}$ (7)
$\displaystyle\sum_{l_{1}\in\boldsymbol{L}}X_{hl_{1}l}(t-1)=\sum_{l_{2}\in\boldsymbol{L}}X_{hl}l_{2}(t),\quad\forall
l\in\boldsymbol{L},\forall h\in\boldsymbol{H},\forall
t\in\boldsymbol{T}\backslash\\{\mathbf{0}\\}$ (8) $\displaystyle
X_{hl_{1}l_{2}}(t)\in\\{0,1\\},E_{h}(\varphi,\omega)\geq
0,F_{oda}(\varphi,\omega)\geq 0,$ $\displaystyle\hskip 179.9pt\forall
a\in\boldsymbol{A},\forall(o,d)\in\boldsymbol{P},\forall
l_{1},l_{2}\in\boldsymbol{L},\forall
h\in\boldsymbol{H},\forall\varphi\in\boldsymbol{T}_{\boldsymbol{t}},\forall\omega\in\boldsymbol{W}_{\boldsymbol{t}},\forall
t\in\boldsymbol{T}$ (9)

The objective function (1) is designed to minimize the expected total costs
over the scenario set during the entire planning horizon. Flow conservation
constraints (2)-(4) ensure the proper routing of each freight shipment from
its origin to its destination. Constraints (5) prevent capacity restriction
violations for each logistic hub. Constraints (6) affirm that the initial
throughput capacity at each logistic hub is empty. Constraints (7) and (8)
assure the selection of exactly one hub capacity deployment strategy in each
planning season, with consecutive changes in capacity levels for each hub.
Lastly, constraints (9) specify the domains of the model variables.

In this study, we tackle two types of uncertainty: demand and travel time
uncertainty. To address computational challenges in stochastic models with
continuous distributions, we adopt a common approach of modeling uncertainty
using discrete probabilistic scenarios. Specifically, we utilize a Poisson
distribution to generate weekly demand from the Freight Analysis Framework
(FAF) yearly demand dataset [16]. The Poisson arrival rate is determined by
the historical monthly demand share, evenly distributed across four weeks
within each month. This process is iterated 50 times to create 50 scenarios,
all sharing the same arrival rate. For travel time uncertainty, we consider
two key parameters: disruption rate and intensity. We assume that hubs operate
independently and have a probability of being disrupted based on a given rate.
For example, a disruption rate of 0.05 and intensity of 1.5 mean that each hub
faces a 5% probability of disruption, and the travel time on the arc
connecting to the interruption hub is 1.5 times the normal travel time. This
simulates scenarios such as construction and maintenance within a hub,
occurring independently.

The problem size, in terms of variables and constraints, significantly expands
with an increasing number of scenarios. Given our focus on a large-scale relay
network, the computational load becomes substantial with a large number of
scenarios. To mitigate this issue, a common approach involves approximating
the original set of scenarios with a smaller representative subset. In this
work, we employ the fast forward selection (FFS) method, which is a widely
used scenario reduction approach introduced by [7]. This FFS method selects
the 10 scenarios out of the original scenario set with the shortest
Kantorovich distance from the unselected set, making it the most
representative of the unselected group, thus reducing computational burden
while preserving solution quality.

##  5 Experimental Results

Figure 2: Nodal map (a), facility network (b) and demand flow (c) for multiple
automotive OEMs in Southeast USA

In this section, we employ a yearly demand dataset for finished automotive
vehicles sourced from the FAF database, with a specific focus on the southeast
US. We conduct experiments focusing on critical uncertainty parameters, aiming
to gain a comprehensive understanding of their influence on the economic and
social performance of the system. The network nodes, as depicted in Figure
2(a), comprise 22 logistic hubs and 22 region centers. Figure 2(b) provides an
overview of the relay network structure, with edges configured to ensure
travel time along each route is limited to 5.5 hours. This design allows
short-haul drivers to complete a round trip within 11 hours, in compliance
with the daily driving hour limit set by US regulations. Lastly, Figure 2(c)
illustrates the yearly demand flow across the southeast US, reflecting the
distribution of freight volume.

To assess the impact of demand and travel time uncertainty on logistic hub
capacity planning, we run the MSHCRP model across three uncertainty levels:
low, intermediate, and high. Each level corresponds to a specific scenario
set, generated using the method outlined in Section 4. Specifically, at the
low uncertainty level, there is no disruption in the network, focusing
exclusively on demand uncertainty. Moving to the intermediate uncertainty
level, we introduce disruption uncertainty, where disruption scenarios are
generated with a disruption rate of 0.05 and a disruption intensity of 1.5.
Lastly, at the high uncertainty level, we examine a scenario with a disruption
rate of 0.05 and a disruption intensity of 15, representing a significant
delay penalty cost. To gain deeper insights into the benefits of dynamic
modeling, we further compare the MSHCRP model with its static counterpart,
where contracted capacities, once determined by the model, remain fixed
throughout the entire planning horizon. All results are summarized in Table 2.

Table 2: Performance Metrics under Different Uncertainty Levels

Uncertainty Levels | Low |  |  | Intermediate |  |  | High |
---|---|---|---|---|---|---|---|---
Model types | Static | Dynamic |  | Static | Dynamic |  | Static | Dynamic
Total Contracted Throughput Capacity | 4,080 | 3,720 |  | 4,200 | 3,870 |  | 4,320 | 3,870
Average Hub Throughput Capacity | 72.86 | 66.43 |  | 70.00 | 64.50 |  | 72.00 | 63.80
Average Number of Hubs | 14.00 | 14.00 |  | 15.00 | 15.00 |  | 15.00 | 15.00
Average Hub Network Connectivity | 4.43 | 4.43 |  | 4.67 | 4.67 |  | 4.67 | 4.52
Hub Costs ($) | 11,166,155 | 9,759,231 |  | 11,281,953 | 9,953,695 |  | 12,108,207 | 10,885,179
Extra Capacity Usage Percentage (%) | 15.19 | 5.67 |  | 13.78 | 5.27 |  | 19.04 | 11.54
Extra Capacity Usage Frequency (%) | 3.53 | 1.65 |  | 3.18 | 1.39 |  | 4.12 | 3.48
Transportation Costs ($) | 34,558,081 | 34,623,956 |  | 35,185,264 | 35,249,483 |  | 36,004,177 | 36,138,348
Disruption Time Percentage (%) | 0.00 | 0.00 |  | 4.77 | 4.92 |  | 4.70 | 5.31
Disrupted Edge Usage Frequency (%) | 0.00 | 0.00 |  | 3.12 | 3.24 |  | 1.23 | 1.40
Fuel Consumption (tons) | 38,878 | 38,952 |  | 39,583 | 39,656 |  | 40,505 | 40,656
CO2 Emissions (metric tons) | 107,924 | 108,129 |  | 109,882 | 110,083 |  | 112,440 | 112,859
Total Costs ($) | 45,724,236 | 44,383,187 |  | 46,467,217 | 45,203,177 |  | 48,112,384 | 47,023,527

Firstly, we focus on the dynamic model (specifically the MSHCRP model) under
the changes in uncertainty levels. As shown in Table 2, the total costs
exhibit an increase from $44,383,187 to $47,023,527 with the escalation of
uncertainty levels. Simultaneously, we can observe a decline in the average
hub throughput capacity and a rise in the number of hubs. This suggests a
trend within the model towards opening more hubs with a smaller design in the
face of larger uncertainty, strategically mitigating hub disruption risks. In
the transportation dimension, the fuel consumption increases from 108,129 tons
to 112,859 tons, indicating the model is inclined to prioritize alternative
longer routes to avoid disrupted arcs, thereby minimizing their impacts on
operations.

Subsequently, we shift the focus to the comparison between the static model
and dynamic model, as outlined in Table 2. The findings reveal that, through
dynamic capacity planning, the carrier can realize savings exceeding
$1,000,000 in hub costs by adjusting their logistic hub capacities across
various planning seasons in response to evolving operational needs. Despite
the reduced contracted capacity, the dynamic model uses approximately one-half
to one-third of extra capacity compared to the static model, showcasing
robustness in dynamic environments. On the other hand, the reduction in total
hub capacity within the dynamic model has a negligible impact on hub network
connectivity, transportation metrics, and environmental metrics, all of which
remain relatively consistent with the static model. Overall, the results
demonstrate that adopting the dynamic setting enables RTCs to proactively
respond to dynamic circumstances, curtail avoidable expenditures, and enhance
comprehensive logistical efficiency.

##  6 Conclusion

This study introduces a two-stage stochastic optimization model to tackle the
multi-period stochastic hub capacity planning with routing problem (MSHCRP) in
the context of relay transportation, which few researchers have paid attention
to. Additionally, we leverage a scenario reduction algorithm based on FFS to
approximate continuous distributions for demand and travel time. In the
experimental section, we analyze the impact of uncertainties across four
dimensions (network structure, hub contract, transportation, and environment)
and compare the model with the static capacity planning model. The results
demonstrate that dynamic capacity planning outperforms static capacity
planning in terms of network robustness and cost-effectiveness, while
maintaining strong performance in delivery efficiency and environmental
impact.

Several promising avenues for future research can be identified. Firstly,
investigating advanced computational methods like accelerated Benders
decomposition and branch-and-price could enhance the solution approach, making
it more efficient for large-scale instances. Secondly, the integration of
relay-based transportation with emerging technologies, such as automatic and
connected trucks, offers an exciting prospect for exploration. Thirdly, a more
detailed categorization of hub capacity into distinct segments, such as
loading zone capacity, parking zone capacity, and driver-switching zone
capacity, could be beneficial. By implementing this approach, one can gain
deeper insights, optimize resource allocation, and develop more effective
strategies tailored to specific operational requirements.

## References

  * [1] ATA, 2021, "Driver shortage update 2021," American Trucking Association (ATA).
  * [2] Li, J., Montreuil, B., and Campos, M., 2022, "Trucker-sensitive hyperconnected large-freight transportation: An operating system," In Proceedings of IISE Annual Conference, 2022.
  * [3] Li, J., Liu, X., and Montreuil, B., 2023, "A multi-agent system for operating hyperconnected freight transportation in the physical internet," IFAC-PapersOnLine, 56(2), 7585–7590.
  * [4] Montreuil, B., 2011, "Toward a physical internet: meeting the global logistics sustainability grand challenge," Logistics Research, 3, 71–87.
  * [5] Kulkarni, O., Dahan, M., and Montreuil, B., 2022, "Resilient hyperconnected parcel delivery network design under disruption risks," International Journal of Production Economics, 251, 108499.
  * [6] Furtado, P., and Frayret, J.M., 2014, "Impact of resource sharing of freight transportation," In 1st International Pysical Internet Conference, Québec City, Canada.
  * [7] Heitsch, H., and Römisch, W., 2003, "Scenario reduction algorithms in stochastic programming," Computational optimization and applications, 24, 187–206.
  * [8] Meller, R. D., Montreuil, B., Thivierge, C., and Montreuil, Z., 2013, "Functional design of Physical Internet facilities: a road-based transit center," CIRRELT Montreal, Canada.
  * [9] Nagy, G., and Salhi, S., 1998, "The many-to-many location-routing problem," Top, 6, 261–275.
  * [10] Alumur, S. A., Campbell, J. F., Contreras, I., Kara, B. Y., Marianov, V., and O’Kelly, M. E., 2021, "Perspectives on modeling hub location problems," European Journal of Operational Research, 291(1), 1–17.
  * [11] Aloullal, A., Saldanha-da Gama, F., and Todosijevi´c, R., 2023, "Multi-period single-allocation hub location-routing: Models and heuristic solutions," European Journal of Operational Research, 310(1), 53–70.
  * [12] Khaleghi, A., and Eydi, A., 2023, "Multi-period hub location problem considering polynomial time-dependent demand," Computers & Operations Research, 159, 106357.
  * [13] Yang, T.-H., 2009, "Stochastic air freight hub location and flight routes planning," Applied Mathematical Modelling, 33(12), 4424–4430.
  * [14] Butun, C., Petrovic, S., and Muyldermans, L., 2021, "The capacitated directed cycle hub location and routing problem under congestion," European Journal of Operational Research, 292(2), 714–734.
  * [15] Chaharsooghi, S., Momayezi, F., and Ghaffarinasab, N., 2017, "An adaptive large neighborhood search heuristic for solving the reliable multiple allocation hub location problem under hub disruptions," International Journal of Industrial Engineering Computations, 8(2), 191–202.
  * [16] Freight Analysis Framework database, The Bureau of Transportation Statistics, https://www.bts.gov/faf.


# Almost periodic distributions and crystalline measures

S.Yu. Favorov  Serhii Favorov,
iiiFaculty of Mathematics and Computer Science, Jagiellonian University,
iii Lojasiewicza 6, 30-348 Krakow, Poland
iii Faculty of Mathematics and Informatics, Karazin’s Kharkiv National
University
iii Svobody sq., 4, Kharkiv, Ukraine 61022<EMAIL_ADDRESS>
> Abstract. Based on the properties of distributions and measures with
> discrete support, we investigate temperate almost periodic distributions on
> the Euclidean space and connection with their Fourier transforms. We also
> study relations between the Fourier transform of almost periodic
> distributions and their Fourier coefficients. The main result of the article
> is the construction of a crystalline measure on the real line, which is
> neither almost periodic distribution, nor a Fourier quasicrystal.
>
> AMS Mathematics Subject Classification: 46F12, 42B10, 42A75
>
> Keywords: temperate distribution, almost periodic distribution, Fourier
> transform, Fourier coefficients, crystalline measure, Fourier quasicrystal

The concept of Fourier quasicrystal was inspired by experimental discovery of
nonperiodic atomic structures with diffraction patterns consisting of spots,
made in the middle of 80’s. A number of papers has appeared, in which the
properties of Fourier quasicrystals are studied. Conditions for support of
quasicrystals to be a finite union of discrete lattices are found, and
nontrivial examples of quasicrystals are constructed
([24],[2],[4]-[6],[9]-[12], [14]-[17],[19]-[22]). These studies have been
extended to the more general setting of temperate distributions with discrete
support and spectrum ([18],[3],[7],[8], [23]). Note that the properties of
almost periodic measures and distributions were more or less explicitly used
in these investigates.

The structure of this article is as follows. In Section 1 we give the
necessary definitions and notations. In Section 2 we prove some properties of
distributions with locally finite support and spectrum. These results are a
slight enhancement of the results of [3] and are included here for the sake of
completeness. In section 3 we introduce a notion of s-almost periodicity of
temperate distributions, which is very close to the classical almost
periodicity. Here we present some theorems on connections between temperate
distributions and their Fourier transform, some of them refer to almost
periodic distributions, and others to s-almost periodic distributions. Based
on them, we obtain the main result of the paper: the example of a crystalline
measure that is not a Fourier quasicrystal. In Section 4 we give proofs of
some theorems from Section 3. Finally, in Section 5 we show that Meyer’s
Theorem on the connection between the Fourier transform of almost periodic
functions and measures and their Fourier coefficients [22] can be generalized
to temperate almost periodic distributions and their Fourier coefficients in
the sense of L.Ronkin [25].

##  1\. Definitions and notations

Denote by $S({\mathbb{R}}^{d})$ the Schwartz space of test functions
$\varphi\in C^{\infty}({\mathbb{R}}^{d})$ with the finite norms

$N_{n,m}(\varphi)=\sup_{{\mathbb{R}}^{d}}\\{\max\\{1,|x|^{n}\\}\max_{\|k\|\leq
m}|D^{k}\varphi(x)|\\},\quad n,m=0,1,2,\dots,$

where

$k=(k_{1},\dots,k_{d})\in({\mathbb{N}}\cup\\{0\\})^{d},\
\|k\|=k_{1}+\dots+k_{d},\
D^{k}=\partial^{k_{1}}_{x_{1}}\dots\partial^{k_{d}}_{x_{d}}.$

These norms generate the topology on $S({\mathbb{R}}^{d})$. Elements of the
space $S^{*}({\mathbb{R}}^{d})$ of continuous linear functionals on
$S({\mathbb{R}}^{d})$ are called temperate distributions. For each temperate
distribution $f$ there are $C<\infty$ and $n,\,m\in{\mathbb{N}}\cup\\{0\\}$
such that for all $\varphi\in S({\mathbb{R}}^{d})$

(1) $|f(\varphi)|\leq CN_{n,m}(\varphi).$

Moreover, this estimate is sufficient for the distribution $f$ to belong to
$S^{*}({\mathbb{R}}^{d})$ (see [27], Ch.3).

The Fourier transform of a temperate distribution $f$ is defined by the
equality

$\hat{f}(\varphi)=f(\hat{\varphi})\quad\mbox{for all}\quad\varphi\in
S({\mathbb{R}}^{d}),$

where

$\hat{\varphi}(y)=\int_{{\mathbb{R}}^{d}}\varphi(x)\exp\\{-2\pi i\langle
x,y\rangle\\}dx$

is the Fourier transform of the function $\varphi$. By $\check{\varphi}$ we
denote the inverse Fourier transform of $\varphi$. The Fourier transform is
the bijection of $S({\mathbb{R}}^{d})$ on itself and the bijection of
$S^{*}({\mathbb{R}}^{d})$ on itself. The support of $\hat{f}$ is called
spectrum of $f$.

We will say that a set $A\subset{\mathbb{R}}^{d}$ is locally finite if the
intersection of $A$ with any ball is finite, $A$ is relatively dense if there
is $R<\infty$ such that $A$ intersects with each ball of radius $R$, and $A$
is uniformly discrete, if $A$ is locally finite and has a strictly positive
separating constant

$\eta(A):=\inf\\{|x-x^{\prime}|:\,x,\,x^{\prime}\in A,\,x\neq x^{\prime}\\}.$

Also, we will say that $A$ is polynomially discrete, or shortly p-discrete, if
there are positive numbers $c,h$ such that

(2) $|x-x^{\prime}|\geq c\min\\{1,\,|x|^{-h},|x^{\prime}|^{-h}\\}\qquad\forall
x,x^{\prime}\in A,\quad x\neq x^{\prime}.$

A set $A$ is of bounded density if it is locally finite and

$\sup_{x\in{\mathbb{R}}^{d}}\\#A\cap B(x,1)<\infty.$

As usual, $\\#E$ is a number of elements of the finite set $E$, and $B(x,r)$
is the ball with center at the point $x$ and radius $r$.

An element $f\in S^{*}({\mathbb{R}}^{d})$ is called a crystalline measure if
$f$ and $\hat{f}$ are complex-valued measures on ${\mathbb{R}}^{d}$ with
locally finite supports.

Denote by $|\mu|(A)$ the variation of the complex-valued measure $\mu$ on $A$.
If both measures $|\mu|$ and $|\hat{\mu}|$ have locally finite supports and
belong to $S^{*}({\mathbb{R}}^{d})$, we say that $\mu$ is a Fourier
quasicrystal. A measure
$\mu=\sum_{\lambda\in\Lambda}a_{\lambda}\delta_{\lambda}$ with
$a_{\lambda}\in{\mathbb{C}}$ and countable $\Lambda$ is called purely point,
here $\delta_{y}$ is the unit mass at the point $y\in{\mathbb{R}}^{d}$. If
this is the case, we will replace $a_{\lambda}$ with $\mu(\lambda)$ and write
$\operatorname{supp}\mu=\\{\lambda:\,\mu(\lambda)\neq 0\\}$.

##  2\. Temperate distributions with locally finite support

By [26], every distribution $f$ with locally finite support $\Lambda$ has the
form

$f=\sum_{\lambda\in\Lambda}P_{\lambda}(D)\delta_{\lambda},\quad
P_{\lambda}(x)=\sum_{\|k\|\leq K_{\lambda}}p_{k}(\lambda)x^{k},\quad
x\in{\mathbb{R}}^{d},\ p_{k}(\lambda)\in{\mathbb{C}},\ K_{\lambda}<\infty.$

Here, as usual, $x^{k}=x_{1}^{k_{1}}\cdots x_{d}^{k_{d}}$. Note that
$\operatorname{ord}f=\sup_{\lambda}\deg P_{\lambda}(x)\leq\infty$. .

######  Proposition 1.

Suppose $f\in S^{*}({\mathbb{R}}^{d})$ has a locally finite support $\Lambda$.
Then

i) $\operatorname{ord}f<\infty$, hence,

(3) $f=\sum_{\lambda\in\Lambda}\sum_{\|k\|\leq
K}p_{k}(\lambda)D^{k}\delta_{\lambda},\quad
k\in({\mathbb{N}}\cup\\{0\\})^{d},\quad K=\operatorname{ord}f;$

in particular, if $f$ has a locally finite spectrum $\Gamma$, then
$\operatorname{ord}\hat{f}<\infty$ and

(4) $\hat{f}=\sum_{\gamma\in\Gamma}\sum_{\|j\|\leq
J}q_{j}(\gamma)D^{j}\delta_{\gamma},\quad
j\in({\mathbb{N}}\cup\\{0\\})^{d},\quad J=\operatorname{ord}\hat{f}.$

ii) If $\Lambda$ is $p$-discrete, then there exist $C,\,T<\infty$ such that
for all $k$

(5) $|p_{k}(\lambda)|\leq C\max\\{1,|\lambda|^{T}\\}\quad\mbox{for all}\
\lambda\in\Lambda.$

Moreover, there exists $T_{1}<\infty$ such that

(6) $\sum_{\lambda\in\Lambda,|\lambda|<R}\sum_{\|k\|\leq
K}|p_{k}(\lambda)|=O(R^{T_{1}})\quad\mbox{as}\quad R\to\infty.$

Proof of Proposition 1. i) Let $\lambda\in\Lambda$ and $\varepsilon\in(0,1)$
be such that

$\inf\\{|\lambda-\lambda^{\prime}|:\,\lambda^{\prime}\in\Lambda,\,\lambda^{\prime}\neq\lambda\\}>\varepsilon.$

Let $\varphi$ be a non-negative function on ${\mathbb{R}}$ such that

(7) $\varphi(|x|)\in C^{\infty}({\mathbb{R}}^{d}),\quad\varphi(|x|)=0\mbox{
for }|x|>1/2,\quad\varphi(|x|)=1\mbox{ for }|x|\leq 1/3.$

Then set

$\varphi_{\lambda,k,\varepsilon}(x)=\frac{(x-\lambda)^{k}}{k!}\varphi\left(\frac{|x-\lambda|}{\varepsilon}\right)\in
S({\mathbb{R}}^{d}),$

where, as usual, $k!=k_{1}!\cdots k_{d}!$. It is easily shown that

$f(\varphi_{\lambda,k,\varepsilon})=(-1)^{\|k\|}p_{k}(\lambda).$

Let $f$ satisfy (1) with some $m,\,n$. We get

$|f(\varphi_{\lambda,k,\varepsilon})|\leq
C\sup_{|x-\lambda|<\varepsilon}\max\\{1,|x|^{n}\\}\sum_{\|\alpha+\beta\|\leq
m}c(\alpha,\beta)\left|D^{\alpha}\varphi\left(\frac{|x-\lambda|}{\varepsilon}\right)D^{\beta}\left(\frac{(x-\lambda)^{k}}{k!}\right)\right|,$

where $\alpha,\beta\in({\mathbb{N}}\cup\\{0\\})^{d}$ and
$c(\alpha,\beta)<\infty$. Note that

$\left|D^{\alpha}\varphi\left(\frac{|x-\lambda|}{\varepsilon}\right)\right|\leq\varepsilon^{-\|\alpha\|}c(\alpha)\quad\mbox{for}\quad|\lambda-x|<\varepsilon/3,$

and this derivative vanishes for $|\lambda-x|\geq\varepsilon/2$. Also,

$D^{\beta}(x-\lambda)^{k}=\begin{cases}0&\text{ if }k_{j}<\beta_{j}\text{ for
at least one }j,\\\ c(k,\beta)(x-\lambda)^{k-\beta}&\text{ if
}k_{j}\geq\beta_{j}\,\ \forall j.\end{cases}.$

Since

$\max\\{1,|x|^{n}\\}\leq 2^{n}\max\\{1,|\lambda|^{n}\\}$

for $x\in\operatorname{supp}\varphi_{\lambda,k,\varepsilon}$, we get

$|p_{k}(\lambda)|\leq\sum_{\|\alpha+\beta\|\leq m,\,\beta_{j}\leq
k_{j}\,\forall
j}c(k,\alpha,\beta)\max\\{1,|\lambda|^{n}\\}\varepsilon^{\|k\|-\|\alpha+\beta\|}.$

For $\|k\|>m$ we take $\varepsilon\to 0$ and obtain $p_{k}(\lambda)=0$.

Since $\hat{f}\in S^{*}({\mathbb{R}}^{d})$, we obtain (4).

ii) Let $\Lambda$ be $p$-discrete and (5) be not satisfy. Then there is
$k,\|k\|\leq K,$ and a sequence $\lambda_{s}\to\infty$ such that
$|\lambda_{s+1}|>1+|\lambda_{s}|$ for all $s$ and

(8) $\log|p_{k}(\lambda_{s})|/\log|\lambda_{s}|\to\infty,\quad s\to\infty.$

Put $\beta_{s}=c|\lambda_{s}|^{-h}$ with $c$ from (2) and

$\psi_{s,k}(x)=\frac{(x-\lambda_{s})^{k}}{k!}\varphi\left(\frac{|x-\lambda_{s}|}{\beta_{s}}\right),\qquad\Psi_{k}(x)=\sum_{s=1}^{\infty}\frac{\psi_{s,k}(x)}{p_{k}(\lambda_{s})}.$

We may suppose that $\lambda_{1}$ is large enough such that
$\operatorname{supp}\psi_{s,k}\cap\operatorname{supp}\psi_{s^{\prime},k}=\emptyset,\
s\neq s^{\prime}$. Then by (8),

$1/p_{k}(\lambda_{s})=o(1/|\lambda_{s}|^{T}),\quad|\lambda_{s}|\to\infty,\quad\mbox{
for every }T<\infty.$

Since

$D^{j}(\psi_{s,k}(x))=O(|\lambda_{s}|^{h\|j\|}),\
j\in({\mathbb{N}}\cup\\{0\\})^{d},$

we see that

$D^{j}(\Psi_{k}(x))=o(1/|x|^{T-h\|j\|}),\quad x\to\infty,$

and $\Psi_{k}\in S({\mathbb{R}}^{d})$.

Since $\Lambda$ is $p$-discrete, we get $\lambda\not\in
B(\lambda_{s},c|\lambda_{s}|^{-h})$ for all
$\lambda\in\Lambda\setminus\\{\lambda_{s}\\}$. Therefore, $f(\Psi_{k})$ is
equal to

$\sum_{\lambda\in\Lambda}\sum_{\|l\|\leq
K}\sum_{s}(-1)^{\|l\|}p_{l}(\lambda)p_{k}(\lambda_{s})^{-1}D^{l}(\psi_{s,k})(\lambda)=\sum_{s}\sum_{\|l\|\leq
K}(-1)^{\|l\|}p_{l}(\lambda_{s})p_{k}(\lambda_{s})^{-1}D^{l}(\psi_{s,k})(\lambda_{s}).$

Since $D^{l}(\psi_{s,k})(\lambda_{s})=0$ for $l\neq k$ and
$D^{k}(\psi_{s,k})(\lambda_{s})=1$, we obtain the contradiction.

Estimate (6) follows immediately from (5) and the following simple lemma:

######  Lemma 1 (cf. [8], a part of the proof of Theorem 6).

If $S$ is $p$-discrete set, then $\\#S\cap B(0,R)=O(R^{T^{\prime}})$ as
$R\to\infty$ with $T^{\prime}<\infty$.

Remark. Proposition 1 was earlier proved by V.Palamodov [23] for temperate
distributions with uniformly discrete support.

######  Proposition 2.

Let $\mu\in S^{*}({\mathbb{R}}^{d})$ be a measure. Then $|\mu|$ belongs to
$S^{*}({\mathbb{R}}^{d})$ if and only if there is $T<\infty$ such that
$|\mu|(B(0,R))=O(R^{T})$ as $R\to\infty$.

Proof. Any non-negative measure $\nu$ on ${\mathbb{R}}^{d}$ satisfying the
condition $\nu(B(0,R))=O(R^{T})$ as $R\to\infty$ belongs to
$S^{*}({\mathbb{R}}^{d})$ (cf.[26]). The converse statement see [6], Lemma 1.

It follows from Propositions 1 and 2

######  Theorem 1 (cf. also [8]).

Let a measure $\mu$ has p-discrete support and belongs to
$S^{*}({\mathbb{R}}^{d})$. Then $|\mu|\in S^{*}({\mathbb{R}}^{d})$ too. In
particular, every crystalline measure with p-discrete support and p-discrete
spectrum is the Fourier quasicrystal.

M.Kolountzakis, J.Lagarias proved in [11] that the Fourier transform of every
measure $\mu$ on the line ${\mathbb{R}}$ with locally finite support of
bounded density, bounded masses $\mu(x)$, and locally finite spectrum is also
a measure $\hat{\mu}=\sum_{\gamma\in\Gamma}q_{\gamma}\delta_{\gamma}$ with
uniformly bounded $q_{\gamma}$. The following proposition generalizes this
result for distributions from $S^{*}({\mathbb{R}}^{d})$.

######  Proposition 3.

Suppose $f\in S^{*}({\mathbb{R}}^{d})$ has form (3) with some $K$ and
countable $\Lambda$, and $\hat{f}$ has form (4) with the locally finite
support $\Gamma$. If

$\rho_{f}(r):=\sum_{|\lambda|<r}\sum_{\|k\|\leq
K}|p_{k}(\lambda)|=O(r^{d+H}),\quad r\to\infty,\quad H\geq 0,$

then $\operatorname{ord}\hat{f}\leq H$; if $\rho_{f}(r)=o(r^{d+H})$ as
$r\to\infty$, then $\operatorname{ord}\hat{f}<H$.

Furthermore, in the case of integer $H$ and $\|j\|=H$ we have
$|q_{j}(\gamma)|\leq C^{\prime}\max\\{1,|\gamma|^{K}\\}$; for the case of
uniformly discrete $\Gamma$ this estimate with the same $K$ takes place for
all $j$.

######  Corollary 1.

If $f\in S^{*}({\mathbb{R}}^{d})$ has form (3) with countable $\Lambda$,
locally finite spectrum $\Gamma$, and $\rho_{f}(r)=O(r^{d})$ as $r\to\infty$,
then $\hat{f}$ is a measure, and

$\hat{f}=\sum_{\gamma\in\Gamma}q(\gamma)\delta_{\gamma},\ |q(\gamma)|\leq
C^{\prime}\max\\{1,|\gamma|^{K}\\}.$

Proof of Proposition 3. Let $\gamma\in\Gamma$ and pick $\varepsilon\in(0,1)$
such that

$\inf\\{|\gamma-\gamma^{\prime}|:\,\gamma^{\prime}\in\Gamma,\,\gamma^{\prime}\neq\gamma\\}>\varepsilon.$

Let $\varphi$ be the same as in the proof of Proposition 1. Put

$\varphi_{\gamma,l,\varepsilon}(y)=\frac{(y-\gamma)^{l}}{l!}\varphi(|y-\gamma|/\varepsilon)\in
S({\mathbb{R}}^{d}).$

We have

$(-1)^{\|l\|}q_{l}(\gamma)=\sum_{\|j\|\leq
J}q_{j}(\gamma)D^{j}\delta_{\gamma}(\varphi_{\gamma,l,\varepsilon}(y))=(\hat{f},\varphi_{\gamma,l,\varepsilon})=(f,\hat{\varphi}_{\gamma,l,\varepsilon}).$

Note that

$\hat{\varphi}_{\gamma,l,\varepsilon}(x)=e^{-2\pi i\langle
x,\gamma\rangle}(l!)^{-1}(-2\pi
i)^{-\|l\|}D^{l}(\widehat{\varphi(\cdot/\varepsilon)})=c(l)e^{-2\pi i\langle
x,\gamma\rangle}\varepsilon^{d+\|l\|}(D^{l}\hat{\varphi})(\varepsilon x).$

Therefore,

$D^{k}(\hat{\varphi}_{\gamma,l,\varepsilon})(x)=\varepsilon^{d+\|l\|}\sum_{\alpha+\beta=k}c(\alpha,\beta)D^{\alpha}\left[e^{-2\pi
i\langle x,\gamma\rangle}\right]D^{\beta}[(D^{l}\hat{\varphi})(\varepsilon
x)]$ $=\sum_{\alpha+\beta=k}c(\alpha,\beta)(-2\pi
i)^{\|\alpha\|}\gamma^{\alpha}e^{-2\pi i\langle
x,\gamma\rangle}\varepsilon^{d+\|l\|+\|\beta\|}(D^{\beta+l}\hat{\varphi})(\varepsilon
x).$

Since $\hat{\varphi}(\varepsilon x)\in S({\mathbb{R}}^{d})$, we get for every
$x\in{\mathbb{R}}^{d}$ and $n\in{\mathbb{N}}\cup\\{0\\}$

$|D^{\beta+l}(\hat{\varphi})(\varepsilon x)|\leq
N_{n,\|\beta+l\|}(\hat{\varphi})(\max\\{1,|\varepsilon x|^{n}\\})^{-1}.$

Therefore for every $k$, $\|k\|\leq K$,

$|D^{k}(\hat{\varphi}_{\gamma,l,\varepsilon})(x)|\leq
C(K,n)\varepsilon^{d+\|l\|}\max\\{1,|\gamma|^{K}\\}(\max\\{1,|\varepsilon
x|^{n}\\})^{-1},$

where $C(k,n)$ depends on $\varphi$. Now we may estimate
$(f,\hat{\varphi}_{\gamma,l,\varepsilon})$ as

(9)
$\left|\sum_{k}\sum_{\lambda}p_{k}(\lambda)D^{k}(\hat{\varphi}_{\gamma,l,\varepsilon})(\lambda)\right|\leq
C(K,n)\varepsilon^{d+\|l\|}\max\\{1,|\gamma|^{K}\\}\int_{0}^{\infty}\frac{\rho_{f}(dt)}{\max\\{1;\,(\varepsilon
t)^{n}\\}}.$

If $\rho_{f}(r)=O(r^{d+H})$ for $r\to\infty$, take $t_{0}$ such that
$|\rho(t)|<C_{0}t^{d+H}$ for $t>t_{0}$. If $\rho_{f}(r)=o(r^{d+H})$, fix any
$\eta>0$ and take $t_{0}=t_{0}(\eta)$ such that $|\rho(t)|<\eta t^{d+H}$ for
$t>t_{0}$. Then pick $n>d+H$ and $\varepsilon<1/t_{0}$. Integrating by parts
and using the estimate for $\rho_{f}(t)$, we obtain

$\int_{0}^{\infty}\max\\{1,(\varepsilon
t)^{n}\\}^{-1}\rho_{f}(dt)=\rho_{f}(1/\varepsilon)+\int_{1/\varepsilon}^{\infty}(\varepsilon
t)^{-n}\rho_{f}(dt)\leq\frac{nC_{0}}{\varepsilon^{n}}\int_{1/\varepsilon}^{\infty}t^{d+H-n-1}dt.$

Therefore, the left-hand side of (9) not more than
$\varepsilon^{\|l\|-H}C_{0}C^{\prime}\max\\{1,|\gamma|^{K}\\}$, and

$|q_{l}(\gamma)|\leq
C^{\prime}C_{0}\max\\{1,|\gamma|^{K}\\}\varepsilon^{\|l\|-H}.$

If $\|l\|>H$, we take $\varepsilon\to 0$ and get $q_{l}(\gamma)=0$, hence,
$J=\operatorname{ord}\hat{f}\leq H$.

If $H$ is integer, we get $|q_{l}(\gamma)|\leq
C^{\prime}C_{0}\max\\{1,|\gamma|^{K}\\}$ for $\|l\|=H$.

If $\rho_{f}(r)=o(r^{d+H})$, we replace $C_{0}$ by $\eta$ and note that $\eta$
is arbitrary small for $\varepsilon$ small enough. Hence, $q_{l}(\gamma)=0$
for $\|l\|=H$.

Finally, if $\Gamma$ is uniformly discrete, we take
$\varepsilon=\varepsilon_{0}<\eta(\Gamma)/2$ for all $\gamma\in\Gamma$ and
obtain the bound

$|q_{l}(\gamma)|\leq\varepsilon_{0}^{-H}C^{\prime}C_{0}\max\\{1,|\gamma|^{K}\\}\quad\forall
l,\|l\|\leq J.$

##  3\. Almost periodic distributions and their properties

Recall that a continuous function $g$ on ${\mathbb{R}}^{d}$ is almost periodic
if for any $\varepsilon>0$ the set of $\varepsilon$-almost periods of $g$

$\\{\tau\in{\mathbb{R}}^{d}:\,\sup_{x\in{\mathbb{R}}^{d}}|g(x+\tau)-g(x)|<\varepsilon\\}$

is a relatively dense set in ${\mathbb{R}}^{d}$.

Almost periodic functions are uniformly bounded on ${\mathbb{R}}^{d}$. The
class of almost periodic functions is closed with respect to taking absolute
values, and finite linear combinations; the limit of a uniformly in
${\mathbb{R}}^{d}$ convergent sequence of almost periodic functions is also
almost periodic.

A typical example of an almost periodic function is every absolutely
convergent exponential sum $\sum c_{n}\exp\\{2\pi i\langle
x,\omega_{n}\rangle\\}$ with
$\omega_{n}\in{\mathbb{R}}^{d},\,c_{n}\in{\mathbb{C}}$ (cf.,for example, [1],
[22]).

A measure $\mu$ on ${\mathbb{R}}^{d}$ is called almost periodic if the
function

$(\psi\star\mu)(t)=\int_{{\mathbb{R}}^{d}}\psi(t-x)d\mu(x)$

is almost periodic in $t\in{\mathbb{R}}^{d}$ for each continuous function
$\psi$ on ${\mathbb{R}}^{d}$ with compact support. A distribution $f\in
S^{*}({\mathbb{R}}^{d})$ is almost periodic if the function $(\psi\star
f)(t)=f(\psi(t-\cdot))$ is almost periodic in $t\in{\mathbb{R}}^{d}$ for each
$\psi\in C^{\infty}$ with compact support (see [13], [25], [19], [20], [22]).
Clearly, every almost periodic distribution has a relatively dense support.
But there are measures that are almost periodic temperate distributions, but
are not almost periodic as measures (see [19]).

######  Definition.

A distribution $f\in S^{*}({\mathbb{R}}^{d})$ is s-almost periodic, if the
function $(\psi\star f)(t)=f(\psi(t-\cdot))$ is almost periodic in
$t\in{\mathbb{R}}^{d}$ for each $\psi\in S({\mathbb{R}}^{d})$.

The following theorem plays a very important role in our investigations.

######  Theorem 2.

If $f$ is a temperate distribution and its Fourier transform $\hat{f}$ is a
pure point measure such that $|\hat{f}|(B(0,r))=O(r^{T})$ for $r\to\infty$
with some $T<\infty$, then $f$ is s-almost periodic distribution.

The proof is very easy. Let
$\hat{f}=\sum_{\gamma\in\Gamma}b(\gamma)\delta_{\gamma}$,
$M(r)=|\hat{f}|(B(0,r))$. For any $\psi\in S({\mathbb{R}}^{d})$ we have

(10) $f(\psi(t-\cdot))=(\hat{f}(y),\hat{\psi}(y)e^{2\pi i\langle
t,y\rangle})=\sum_{\gamma\in\Gamma}b(\gamma)\hat{\psi}(\gamma)e^{2\pi i\langle
t,\gamma\rangle}.$

Since $|\hat{\psi}(y)|\leq N_{T-1,0}(\hat{\psi})|y|^{-T-1}$ for $|y|>1$ and

$\sum_{\gamma\in\Gamma}|b(\gamma)||\hat{\psi}(\gamma)|\leq
C_{0}+C_{1}\int_{1}^{\infty}r^{-T-1}M(dr)<\infty,$

we see that the series in (10) absolutely converges, and the function
$(f\star\psi)(t)$ is almost periodic.

From Proposition 2 it follows

######  Corollary 2.

If $f\in S^{*}({\mathbb{R}}^{d})$, $\hat{f}$ is pure point measure, and
$|\hat{f}|\in S^{*}({\mathbb{R}}^{d})$, then $f$ is s-almost periodic
distribution. In particular, every Fourier quasicrystal is s-almost periodic
distribution.

Using Proposition 1, we also get

######  Corollary 3.

Let $f\in S^{*}({\mathbb{R}}^{d})$ have $p$-discrete spectrum, and the Fourier
transform $\hat{f}$ is a measure. Then $f$ is s-almost periodic distribution.

######  Corollary 4.

Let $f\in S^{*}({\mathbb{R}}^{d})$ of form (3) have locally finite spectrum
$\Gamma$ with polynomial growth of numbers $\\#(\Gamma\cap B(0,r))$. If

$\sum_{\|k\|\leq
K}\sum_{|\lambda|<r}|p_{k}(\lambda)|=O(r^{d})\quad\mbox{for}\quad r\to\infty,$

then $f$ is s-almost periodic distribution.

Indeed, by Corollary 1, $\hat{f}$ is a measure with polynomially growth
coefficients.

Evidently, every s-almost periodic distribution is an almost periodic
distribution too. I do not know if there is an almost periodic distribution,
which is not s-almost periodic. However, the following assertion is valid:

######  Theorem 3.

Every almost periodic (in sense of distributions) non-negative measure $\mu\in
S^{*}({\mathbb{R}}^{d})$ is s-almost periodic distribution. The same
implication is valid if $\mu$ is a complex-valued measure on
${\mathbb{R}}^{d}$ such that

(11) $\sup_{x\in{\mathbb{R}}^{d}}|\mu|(B(x,1))<\infty.$

It is easy to check that every almost periodic in the sense of distributions
measure $\mu$ under condition (11) is almost periodic in sense of measures.

Proofs of Theorem 3 and the following ones are given in the next Section 4.

######  Theorem 4.

If $f\in S^{*}({\mathbb{R}}^{d})$ is an almost periodic distribution with
locally finite spectrum $\Gamma$, then $\hat{f}$ is a measure.

Show that $p$-discreteness of support of a measure is closely connected with
s-almost periodicity of its Fourier transform:

######  Theorem 5.

In order for each measure $\mu\in S^{*}({\mathbb{R}}^{d})$ with support in a
fixed locally finite set $A\subset{\mathbb{R}}^{d}$ to have s-almost periodic
Fourier transform $\hat{\mu}$, it is necessary and sufficient that $A$ be
$p$-discrete.

Moreover, if $\hat{\mu}\star\psi(t)$ is bounded for all $\psi\in
S({\mathbb{R}}^{d})$ and $\mu\in S^{*}({\mathbb{R}}^{d})$ with
$\operatorname{supp}\mu\subset A$, then $A$ is $p$-discrete too.

######  Theorem 6.

There is a crystalline measure $\mu$ on ${\mathbb{R}}$ such that for some
$\psi\in S({\mathbb{R}})$ the function $(\mu\star\psi)(t)$ is unbounded in
$t\in{\mathbb{R}}$. In particular, $\mu$ is neither s-almost periodic
distribution, nor the Fourier quasicrystal.

Remark. Y.Meyer formulated in [20] as a theorem that any crystalline measure
is an almost periodic distribution. Then he wrote in [21] that the proof of
this theorem is incorrect and formulated the corresponding result as
Conjecture 2.1. Theorem 6 gives only a partial answer on this conjecture,
because the function $\psi\in S({\mathbb{R}})$, which constructs in the proof
of Theorem 6, does not have compact support. We don’t know if there is a
function $\tilde{\psi}\in C^{\infty}$ with compact support such that
$(\mu\star\tilde{\psi})(t)$ is unbounded.

##  4\. Proofs of the theorems

Proof of Theorem 3. Let $\varphi$ be $C^{\infty}$ non-negative function with
compact support such that $\varphi(x)\equiv 1$ for $x\in B(0,1)$. Since
$\varphi\star\mu(t)$ is an almost periodic function, we see that it is
uniformly bounded. If $\mu\geq 0$, we get $\mu(B(x,1))<C$ for all
$x\in{\mathbb{R}}^{d}$.

Set $\mu^{t}(x):=\mu(t-x)$ with $t\in{\mathbb{R}}^{d}$. For every complex-
valued measure $\mu$ under condition (11) we get
$M(r):=|\mu^{t}|(B(0,r))<Cr^{d}$ for all $r>1$, where the constant $C$ is the
same for all $t$. Take $\psi\in S({\mathbb{R}}^{d})$. Then $|\psi(x)|\leq
C_{1}|x|^{-d-1}$ for $|x|>1$. For any $\varepsilon>0$ there is $R<\infty$ that
does not depend on $t$ and such that

$\left|\int_{|x|>R}\psi(x)\mu^{t}(dx)\right|\leq
C_{1}\int_{R}^{\infty}r^{-d-1}M(dr)\leq
C_{1}(d+1)\int_{R}^{\infty}M(r)r^{-d-2}dr<\varepsilon/3.$

Therefore for all $t\in{\mathbb{R}}^{d}$

(12)
$\left|\int_{|t-x|>R}\psi(t-x)\mu(dx)\right|=\left|\int_{|x^{\prime}|>R}\psi(x^{\prime})\mu^{t}(dx^{\prime})\right|<\varepsilon/3.$

Let $\xi(x)$ be $C^{\infty}$-function on ${\mathbb{R}}^{d}$ such that

$0\leq\xi\leq 1,\quad\xi(x)\equiv 1\quad\text{for}\quad|x|<R,\quad\xi(x)\equiv
0\quad\text{for}\quad|x|>R+1.$

The function $(\xi\psi)\star\mu(t)$ is almost periodic, hence there are a
relatively dense set $E\subset{\mathbb{R}}^{d}$ such that for any $\tau\in E$
and all $t\in{\mathbb{R}}^{d}$

$|(\xi\psi)\star\mu(t+\tau)-(\xi\psi)\star\mu(t)|<\varepsilon/3.$

Applying (12) to $(1-\xi(t+\tau))\psi(t+\tau)$ and $(1-\xi(t))\psi(t)$, we
obtain

$|\psi\star\mu(t+\tau)-\psi\star\mu(t)|\leq|(\xi\psi)\star\mu(t+\tau)-(\xi\psi)\star\mu(t)|+|(1-\xi)\psi\star\mu(t+\tau)|+|(1-\xi)\psi\star\mu(t)|<\varepsilon.$

Hence, $E$ is the set of $\varepsilon$-almost periods for the function
$\psi\star\mu$.

Proof of Theorem 4. Let $f$ be an almost periodic temperate distribution with
a locally finite spectrum $\Gamma$. By Proposition 1, $\hat{f}$ has form (4).
Suppose that $J\neq 0$ and $q_{j^{\prime}}(\gamma^{\prime})\neq 0$ for some
$\gamma^{\prime}\in\Gamma$ and
$j^{\prime}=(j^{\prime}_{1},\dots,j^{\prime}_{d}),\,\|j^{\prime}\|=J$. Without
loss of generality suppose that $j^{\prime}_{1}\neq 0$. Set

(13) $e_{1}=(1,0,\dots,0),\quad e_{2}=(0,1,\dots,0),\dots,\
e_{d}=(0,\dots,0,1),$

and $j^{\prime\prime}=j^{\prime}-e_{1}$. Pick
$\varepsilon<\min\\{|\gamma^{\prime}-\gamma|:\,\gamma\in\Gamma\\}$, and set

$\varphi_{\gamma^{\prime},j^{\prime\prime},\varepsilon}(y)=\frac{(y-\gamma^{\prime})^{j^{\prime\prime}}}{j^{\prime\prime}!}\varphi\left(\frac{|y-\gamma^{\prime}|}{\varepsilon}\right),$

where $\varphi$ is defined in (7). We have

(14) $\hat{f}(e^{2\pi i\langle
y,t\rangle}\varphi_{\gamma^{\prime},j^{\prime\prime},\varepsilon}(y))=\sum_{\gamma\in\Gamma}\sum_{\|j\|\leq
J}(-1)^{\|j\|}q_{j}(\gamma)D^{j}(e^{2\pi i\langle
y,t\rangle}\varphi_{\gamma^{\prime},j^{\prime\prime},\varepsilon}(y))(\gamma)$

Since

$D^{j}(\varphi_{\gamma^{\prime},j^{\prime\prime},\varepsilon}(y))(\gamma)=\left\\{\begin{array}[]{l}0\mbox{
if }\gamma\neq\gamma^{\prime}\mbox{ or }j\neq j^{\prime\prime},\\\ 1\mbox{ if
}\gamma=\gamma^{\prime}\mbox{ and }j=j^{\prime\prime},\end{array}\right.$

we see that expression (14) is equal to

$(-1)^{J}q_{j^{\prime}}(\gamma^{\prime})2\pi it_{1}e^{2\pi
i\langle\gamma^{\prime},t\rangle}+(-1)^{J}\sum_{s=2}^{d}q_{j^{\prime\prime}+e_{s}}(\gamma^{\prime})2\pi
it_{s}e^{2\pi
i\langle\gamma^{\prime},t\rangle}+(-1)^{J-1}q_{j^{\prime\prime}}(\gamma^{\prime})e^{2\pi
i\langle\gamma^{\prime},t\rangle}.$

The first summand is unbounded in $t_{1}\in{\mathbb{R}}$, hence the function

$f(\hat{\varphi}_{\gamma^{\prime},j^{\prime\prime},\varepsilon}(x-t))=\hat{f}(e^{2\pi
i\langle
y,t\rangle}\varphi_{\gamma^{\prime},j^{\prime\prime},\varepsilon}(-y))$

is unbounded and not almost periodic. We obtain the contradiction, therefore,
$J=0$ and $\hat{f}$ is a measure.

In order to prove Theorems 5 and 6, we need the following proposition:

######  Proposition 4.

Let $\lambda_{n},\tau_{n}\in{\mathbb{R}}^{d}$ be two sequences such that
$\tau_{n}\to 0$, $|\lambda_{n}|>|\lambda_{n-1}|+1$ for all $n$, and

(15) $\log|\tau_{n}|/\log|\lambda_{n}|\to-\infty\quad\mbox{as}\quad
n\to\infty.$

Let $\mu$ be any measure from $S^{*}({\mathbb{R}}^{d})$ such that its
restriction for each ball $B(\lambda_{n},1/(2|\lambda_{n}|))$ equals
$|\tau_{n}|^{-2/3}(\delta_{\lambda_{n}+\tau_{n}}-\delta_{\lambda_{n}})$. Then
there is $\psi\in S({\mathbb{R}}^{d})$ such that $\hat{\mu}\star\hat{\psi}(t)$
is unbounded. In particular, $\hat{\mu}$ is not s-almost periodic
distribution.

Proof. By thinning out the sequence $\tau_{n}$, we can assume that for all $n$

(16) $\sum_{p<n}|\tau_{p}|^{-1/3}<(1/3)|\tau_{n}|^{-1/3},$

and

(17) $\sum_{p>n}|\tau_{p}|^{2/3}<(2/(3\pi))|\tau_{n}|^{2/3}.$

Set

$\psi(x)=\sum_{n}|\tau_{n}|^{1/3}\varphi(|\lambda_{n}||x-\lambda_{n}|),$

where $\varphi$ is defined in (7). By (15),
$|\tau_{n}|=o(1/|\lambda_{n}|^{T})$ as $n\to\infty$ for every $T<\infty$.
Therefore, for all $k\in({\mathbb{N}}\cup\\{0\\})^{d},\ N\in{\mathbb{N}}$ we
have $D^{k}\psi(x)=o(|\lambda_{n}|^{-N})$ for $x\in
B(\lambda_{n},1/(2|\lambda_{n}|))$. Hence, $(D^{k}\psi)(x)(1+|x|^{N})$ is
bounded on ${\mathbb{R}}^{d}$ for all $N$ and $k$, i.e., $\psi\in
S({\mathbb{R}}^{d})$. By (7), $\psi(x)=0$ for
$x\not\in\cup_{n}B(\lambda_{n},1/(2\lambda_{n}))$. Hence, for every
$t\in{\mathbb{R}}^{d}$

$\hat{\mu}(\hat{\psi}(t-y))=\mu(\psi(x)e^{-2\pi i\langle
x,t\rangle})=\sum_{n=1}^{\infty}|\tau_{n}|^{-1/3}[\varphi(|\tau_{n}||\lambda_{n}|)e^{-2\pi
i\langle(\lambda_{n}+\tau_{n}),t\rangle}-\varphi(0)e^{-2\pi
i\langle\lambda_{n},t\rangle}].$

For large $n$ we have $|\tau_{n}|<1/(3|\lambda_{n}|)$, therefore,
$\varphi(|\tau_{n}||\lambda_{n}|)=\varphi(0)=1$. Besides, for
$t=\tau_{n}/(2|\tau_{n}|^{2})$

$|e^{-2\pi i\langle(\lambda_{n}+\tau_{n}),t\rangle}-e^{-2\pi
i\langle\lambda_{n},t\rangle}|=|e^{-2\pi i\langle\tau_{n},t\rangle}-1|=2.$

Therefore,

(18) $|\hat{\mu}(\hat{\psi}(t-y))|\geq
2|\tau_{n}|^{-1/3}-2\sum_{p<n}|\tau_{p}|^{-1/3}-\sum_{p>n}|\tau_{p}|^{-1/3}|e^{-2\pi
i\langle\tau_{p},t\rangle}-1|.$

Taking into account (16), (17), and the estmates

$|e^{-2\pi i\langle\tau_{p},t\rangle}-1|\leq
2\pi|\tau_{p}||t|=\pi|\tau_{p}||\tau_{n}|^{-1},$

we obtain that (18) is more than $2|\tau_{n}|^{-1/3}/3$. Hence the convolution
$(\hat{\mu}\star\hat{\psi})(t)$ is unbounded on the sequence
$\tau_{n}/(2|\tau_{n}|^{2})$, and the distribution $\hat{\mu}$ is not s-almost
periodic.

Proof of Theorem 5. Suppose that $A$ is not $p$-discrete set. Then there are
two sequences $\lambda_{n},\lambda^{\prime}_{n}\in A$ such that $\lambda_{n}$
and $\tau_{n}:=\lambda^{\prime}_{n}-\lambda_{n}$ satisfy (15) and
$|\lambda_{n}|>1+|\lambda_{n-1}|$. Check that the measure

$\mu=\sum_{n}|\tau_{n}|^{-2/3}[\delta_{\lambda^{\prime}_{n}}-\delta_{\lambda_{n}}]$

belongs to $S^{*}({\mathbb{R}}^{d})$.

For any $\phi\in S({\mathbb{R}}^{d})$ we have

$|(\mu,\phi)|\leq\sum_{n}|\tau_{n}|^{-2/3}|\phi(\lambda^{\prime}_{n})-\phi(\lambda_{n})|\leq\sum_{n}|\tau_{n}|^{1/3}N_{0,1}(\phi),$

where $N_{0,1}(\phi)$ is defined in (1). By (15), $\tau_{n}=O(n^{-T})$ for any
$T<\infty$, therefore the sum converges, $\mu$ satisfies (1), and $\mu$ is a
temperate distribution. Applying Proposition 4, we obtain that $\hat{\mu}$ is
not s-almost periodic.

Proposition 4 actually implies the unboundedness of the convolution
$\hat{\mu}\star\hat{\psi}$ with some $\psi\in S({\mathbb{R}}^{d})$, which
proves the last part of the theorem.

Sufficiency follows from Corollary 3.

Our proof of Theorem 6 uses the following lemmas:

######  Lemma 2 (Y.Meyer [20], Lemma 7).

Let $\alpha\in(0,1/6)$. For every integer $M>M_{\alpha}$ there exists an
$M$-periodic locally finite measure $\sigma=\sigma_{M}$ that is a sum of Dirac
masses on $\Lambda_{M}=M^{-1}{\mathbb{Z}}\setminus[-\alpha M,\alpha M]$, and
whose Fourier transform is also supported by $\Lambda_{M}$. To be precise,

$\operatorname{supp}\sigma=M^{-1}{\mathbb{Z}}\setminus\left[\cup_{k\in{\mathbb{Z}}}(kM+(-\alpha
M,\alpha M)\right].$

For $\tau\in{\mathbb{R}}$ set
$\sigma_{M}^{\tau}=\sum_{\lambda\in\operatorname{supp}\sigma_{M}}\delta_{\lambda+\tau}$.

######  Lemma 3.

Let $M\geq 16,\,\alpha=1/8$. Then for any $\phi\in S({\mathbb{R}})$ and
$\tau\in(0,1/2)$ we have

(19) $|(\sigma_{M}^{2\tau}-\sigma_{M}^{\tau},\phi)|\leq C^{\prime}\tau
N_{2,1}(\phi),$

where $N_{2,1}(\phi)$ is defined in (1), and $C^{\prime}$ is an absolute
constant.

Proof of Lemma 3. Taking into account that $|\lambda|\geq\alpha M>2$ for every
$\lambda\in\operatorname{supp}\sigma_{M}$, we get

$|((\delta_{\lambda+2\tau}-\delta_{\lambda+\tau}),\phi)|\leq\max_{0\leq\theta\leq
1}|\phi^{\prime}(\lambda+(1+\theta)\tau)|\tau\leq\tau
N_{2,1}(\phi)|\lambda|^{-2}.$

Note that a number of points of
$\operatorname{supp}\sigma_{M}^{2\tau}\cup\operatorname{supp}\sigma_{M}^{\tau}$
on the interval

$((k-1)M+(1/8)M,kM-(1/8)M),\,k\geq 1,$

is less than $2M^{2}$. Therefore,

$|(\sigma_{M}^{2\tau}-\sigma_{M}^{\tau},\phi)|\leq 2M^{2}\tau
N_{2,1}(\phi)\sum_{k=1}^{\infty}[(k-1)M+M/8]^{-2}.$

Proof of Theorem 6. Let $\tau_{n}$ be a sequence of mutually independent over
${\mathbb{Z}}$ positive numbers such that $\tau_{n}<1/16$ and
$\log\tau_{n}/n\to-\infty$ as $n\to\infty$. Let $\Lambda_{M_{n}}$ be the set
defined in Lemma 2 with $M_{n}=16^{n},\ \alpha=1/8$, and $\sigma_{M_{n}}$ be
the corresponding measure. It is not hard to check that the sets
$\Lambda_{M_{n}}+\tau_{n},\Lambda_{M_{p}}+2\tau_{p},\ n,p\in{\mathbb{N}},$ are
mutually disjoint if $n,\,p\geq n_{0}$ and $n_{0}$ large enough. Since
$\Lambda_{M_{p}}\cap(-\alpha M_{n},\alpha M_{n})=\emptyset$ for $p\geq n$, we
see that the measure

$\nu=\sum_{n\geq
n_{0}}\tau_{n}^{-2/3}(\sigma_{M_{n}}^{2\tau_{n}}-\sigma_{M_{n}}^{\tau_{n}})$

is locally finite for appropriate $n_{0}$. Using Lemma 3, we obtain

$|(\nu,\phi)|\leq\sum_{n\geq n_{0}}C^{\prime}\tau_{n}^{1/3}N_{2,1}(\phi).$

Since $\tau_{n}\leq n^{-6}$ for $n$ large enough, we see that the sum
converges, the measure $\nu$ satisfies condition (1), and $\nu\in
S^{*}({\mathbb{R}})$. Moreover, its spectrum is a subset of the locally finite
set $\cup_{n\geq n_{0}}\Lambda_{M_{n}}$, therefore the measure
$\mu=\check{\nu}$ is a crystalline measure. By Proposition 4, to check that
$\mu$ is not s-almost periodic distribution we only need to find a sequence
$\lambda_{n}\to\infty$ such that $\log\lambda_{n}=O(n)$ and

$(\lambda_{n}-1/(2\lambda_{n}),\lambda_{n}+1/(2\lambda_{n}))\cap\operatorname{supp}\nu=\\{\lambda_{n},\lambda_{n}+\tau_{n}\\}.$

Fix $n\geq n_{0}$. Set $\eta_{n,j}=M_{n}+j/M_{n}$ with $j\in{\mathbb{N}}$ such
that $M_{n}+M_{n}/8\leq\eta_{n,j}\leq 2M_{n}-M_{n}/8$, and set
$I_{n,j}:=(\eta_{n,j}-1/(2\eta_{n,j}),\eta_{n,j}+1/(2\eta_{n,j}))$. For every
fixed $n$ these intervals do not intersect. Since
$2M_{n}-M_{n}/8+2\tau_{n}<M_{p}/8$ for $p>n$, we get

$I_{n,j}\cap[\operatorname{supp}\sigma_{M_{p}}^{2\tau_{p}}\cup\operatorname{supp}\sigma_{M_{p}}^{\tau_{p}}]=\emptyset\quad\forall
j,\,\forall p>n.$

Then for every $p<n$ a number of points of the form $kM_{p}+q/M_{p}+\tau_{p}$
or $kM_{p}+q/M_{p}+2\tau_{p}$, $k,q\in{\mathbb{N}}\cup\\{0\\}$ on the interval
$(M_{n},2M_{n})$ is at most $2M_{n}M_{p}$. Summing over all $p<n$, we get

$\\#\\{\cup_{p<n}[\operatorname{supp}\sigma_{M_{p}}^{2\tau_{p}}\cup\operatorname{supp}\sigma_{M_{p}}^{\tau_{p}}]\\}\cap(M_{n},2M_{n})<2M_{n}^{2}/15.$

On the other hand, the number of points $\eta_{n,j}$ on the interval
$(M_{n}+M_{n}/8,\,2M_{n}-M_{n}/8)$ is $3M_{n}^{2}/4$, and the same is the
number of intervals $I_{n,j}\subset(M_{n},2M_{n})$. Hence there is
$j^{\prime}$ such that

$I_{n,j^{\prime}}\cap[\operatorname{supp}\sigma_{M_{p}}^{2\tau_{p}}\cup\operatorname{supp}\sigma_{M_{p}}^{\tau_{p}}]=\emptyset\quad\forall
p\neq n.$

Set $\lambda_{n}=\eta_{n,j^{\prime}}+\tau_{n}$. Clearly,
$(16)^{n}<\lambda_{n}<2(16)^{n}$. Since $\tau_{n}<4^{-1}16^{-n}$ for $n$ large
enough, we see that

$\lambda_{n},\,\lambda_{n}+\tau_{n}\in(\lambda_{n}-1/(2\lambda_{n}),\,\lambda_{n}+1/(2\lambda_{n}))\subset
I_{n,j^{\prime}},$

and

$\eta_{j,n}+\tau_{n},\,\eta_{j,n}+2\tau_{n}\not\in(\lambda_{n}-1/(2\lambda_{n}),\,\lambda_{n}+1/(2\lambda_{n}))\quad\mbox{for}\quad
j\neq j^{\prime}.$

We obtain that the measure $\nu$ satisfies Proposition 4 for $n_{0}$ large
enough. It follows from Corollary 2 that the measure $|\hat{\nu}|$ does not
belong to $S^{*}({\mathbb{R}}^{d})$, hence $\hat{\nu}$ is not the Fourier
quasicrystal.

##  5\. Fourier coefficients and Fourier transform of almost periodic
distributions

For any almost periodic function $g$ on ${\mathbb{R}}^{d}$ its coefficient
Fourier for an exponent $\lambda\in{\mathbb{R}}^{d}$ is defined by the formula

$a(\lambda,g)=\lim_{R\to\infty}\frac{1}{\omega_{d}R^{d}}\int_{B(0,R)}g(t)e^{-2\pi
i\langle\lambda,t\rangle}dt,$

where $\omega_{d}$ is the volume of the unit ball in ${\mathbb{R}}^{d}$, and
the limit exists uniformly with respect to shifts of $g$.

Now let $f$ be an almost periodic temperate distribution on
${\mathbb{R}}^{d}$. By L.Ronkin [25], its Fourier coefficient corresponding to
an exponent $\lambda\in{\mathbb{R}}^{d}$ is defined by the equality

$a(\lambda,f)=a(\lambda,f\star\phi)/\hat{\phi}(\lambda),$

where $\phi$ is any $C^{\infty}$-function with compact support such that
$\hat{\phi}(\lambda)\neq 0$, and $a(\lambda,f\star\phi)$ is the Fourier
coefficient of the almost periodic function $f\star\phi$. Note that the
definition matches with the previous one for an almost periodic function $g$.
Indeed, in this case

$a(\lambda,g\star\phi)=\lim_{R\to\infty}\frac{1}{\omega_{d}R^{d}}\int_{B(0,R)}\int_{\operatorname{supp}\phi}g(t-x)\phi(x)e^{-2\pi
i\langle\lambda,t\rangle}dx\,dt$
$=\int_{\operatorname{supp}\phi}\left[\lim_{R\to\infty}\frac{1}{\omega_{d}R^{d}}\int_{B(0,R)}g(t-x)e^{-2\pi
i\langle\lambda,(t-x)\rangle}dt\right]\phi(x)e^{-2\pi
i\langle\lambda,x\rangle}dx=a(\lambda,g)\hat{\phi}(\lambda).$

Then for any almost periodic temperate distribution $f$ and any
$\phi,\,\psi\in C^{\infty}({\mathbb{R}}^{d})$ with compact supports such that
$\hat{\phi}(\lambda)\neq 0,\,\hat{\psi}(\lambda)\neq 0$ we apply the above
equality for almost periodic functions $f\star\phi,\,f\star\psi$ and get

$\frac{a(\lambda,f\star\phi)}{\hat{\phi}(\lambda)}=\frac{a(\lambda,(f\star\phi)\star\psi)}{\hat{\phi}(\lambda)\hat{\psi}(\lambda)}=\frac{a(\lambda,(f\star\psi)\star\phi))}{\hat{\phi}(\lambda)\hat{\psi}(\lambda)}=\frac{a(\lambda,f\star\psi)}{\hat{\psi}(\lambda)}.$

Therefore the definition of $a(\lambda,f)$ does not depend on a function
$\phi$.

Y.Meyer ([22], Theorem 3.8) proved that for any almost periodic function $g$
the following conditions are equivalent

i) the sum $\sum_{|\lambda|<R}|a(\lambda,g)|$ converges for every $R<\infty$,

ii) $\hat{g}$ is a measure,

iii) $\hat{g}$ is an atomic measure of the form
$\hat{g}=\sum_{\lambda}a(\lambda,g)\delta_{\lambda}$.

Then Y.Meyer generalized this result to almost periodic measures.

The result has a simple generalization to almost periodic distributions and
their Fourier coefficients in sense of Ronkin:

######  Theorem 7.

Let $f$ be an almost periodic distribution. Then the following conditions are
equivalent:

i) the sum $\sum_{|\lambda|<R}|a(\lambda,f)|$ converges for every $R<\infty$,

ii) $\hat{f}$ is a measure,

iii) $\hat{f}$ is an atomic measure of the form
$\hat{f}=\sum_{\lambda}a(\lambda,f)\delta_{\lambda}$.

Proof. Check the implication i)$\Rightarrow$iii). Then for any function
$\phi\in C^{\infty}$ with compact support the sum

$\sum_{|\lambda|<R}|a(\lambda,f\star\phi)|=\sum_{|\lambda|<R}|a(\lambda,f)||\hat{\phi}(\lambda)|$

converges for any $R<\infty$. Therefore
$\widehat{f\star\phi}=\hat{f}\hat{\phi}$ is a measure of the form

$\sum_{\lambda}a(\lambda,f\star\phi)\delta_{\lambda}=\sum_{\lambda}a(\lambda,f)\hat{\phi}(\lambda)\delta_{\lambda}.$

Note that any function $\psi\in S({\mathbb{R}}^{d})$ can be approximated by
$C^{\infty}$-functions with compact supports. Therefore for every $\psi\in
S({\mathbb{R}}^{d})$ the sum

$\hat{f}\hat{\psi}=\sum_{\lambda}a(\lambda,f)\hat{\psi}(\lambda)\delta_{\lambda}$

is a continuous linear functional on the space of all continuous functions on
every ball $\overline{B(0,R)},\,R<\infty$. If we take $\psi$ such that
$\hat{\psi}(\lambda)\equiv 1$ for $|\lambda|\leq R$, we get that the
restriction of $\hat{f}$ on $\overline{B(0,R)}$ coincides with
$\sum_{\lambda}a(\lambda,f)\delta_{\lambda}$, and iii) is proved.

There is no need to prove the implication iii)$\Rightarrow$ii).

In order to check the implication ii)$\Rightarrow$i), suppose that $\hat{f}$
is a measure and $\phi$ is any $C^{\infty}$-function with compact support.
Then $\widehat{f\star\phi}=\hat{f}\hat{\phi}$ is also a measure. Since
$f\star\phi$ is an almost periodic function, we see, by Meyer’s Theorem, that
the sum

$\sum_{|\lambda|<R}|a(\lambda,f)||\hat{\phi}(\lambda)|=\sum_{|\lambda|<R}|a(\lambda,f\star\phi)|$

converges for any $R<\infty$. If $\hat{\phi}(\lambda)\neq 0$ for all
$\lambda\in\overline{B(0,R)}$, we obtain i).

I want to thank the referee for a careful reading of my article and important
remarks that forced the author to completely change Theorem 3. I also would
like to thank N.Lev, G.Reti for the inaccuracy they discovered in one
important definition. Finally, I thank the Department of Mathematics and
Computer Science of the Jagiellonian University for their hospitality and
Professor Lukasz Kosinski for his interest in my work and useful discussions.

## References

  * [1] C.Corduneanu, Almost Periodic Functions, Second English ed. Chelsea, New-York, 1989 (Distributed by AMS and Oxford University Press).
  * [2] S.Yu. Favorov. Fourier quasicrystals and Lagarias’ conjecture. Proc. Amer. Math. Soc. 144 (2016) , 3527-3536.
  * [3] S.Yu. Favorov. Tempered distributions with discrete support and spectrum. Bulletin of the Hellenic Mathematical Society, v.62, (2018), 66-79.
  * [4] S.Yu. Favorov. Large Fourier quasicryals and Wiener’s Theorem. Journal of Fourier Analysis and Applications, Vol. 25, Issue 2, (2019), 377-392
  * [5] S.Yu. Favorov. Local Wiener’s Theorem and Coherent Sets of Frequencies. Analysis Math., 46 (4) (2020), 737–746 DOI: 10.1007/s10476-020-0042-x
  * [6] S.Yu. Favorov. Uniqueness Theorems for Fourier Quasicrystals and Temperate Distributions with Discrete Support. Proc. Amer. Math. Soc. 149 (2021), 4431-4440
  * [7] S.Yu. Favorov. Temperate distributions with locally finite support and spectrum on Euclidean spaces. Israel Journal of Mathematics TBD (2022), 1–24.
  * [8] S.Yu. Favorov. Fourier quasicrystals and distributions on Euclidean spaces with spectrum of bounded density. To appear in Analysis Math.
  * [9] M.N.Kolountzakis. On the Structure of Multiple Translations Tilings by Polygonal Regions, Preprint, 1999, 16p.
  * [10] M.N.Kolountzakis. Fourier Pairs of Discrete Support with Little Structure. February 2015 Journal of Fourier Analysis and Applications, 22 no.1, (2016) 1-5.
  * [11] M.N.Kolountzakis, J.C.Lagarias, Structure of Tilings of the Line by a Function. Duke Math.Journal, 82, (1996), 653-678.
  * [12] J.C. Lagarias, Geometric Models for Quasicrystals I.Delone Set of Finite Type. Discr.and Comp.Geometry, 21 161-191 (1999)
  * [13] J.G.de Lamadrid, L.N.Argabright, Almost Periodic Measures. Memoirs of the AMS, No.428, Providence RI, (1990), 218p.
  * [14] N.Lev, A.Olevskii, Measures with Uniformly Discrete Support and Spectrum. C.R.Acad.Sci.,ser.1 351, (2013) 599-603.
  * [15] N.Lev, A.Olevskii, Quasicrystals and Poisson’s Summation Formula. Invent.Math. 200, (2015) 585–606.
  * [16] N.Lev, A.Olevskii, Quasicrystals with Discrete Support and Spectrum. Rev.Mat.Iberoam., 32, no.4, (2016) 1341-1252.
  * [17] N.Lev, A.Olevskii, Fourier Quasicrystals and Discreteness of the Diffraction Spectrum. Advances in Mathematics, 315, (2017) 1-26.
  * [18] N.Lev and G.Reti Crystalline Temperate Distribution with Uniformly Discrete Support and Spectrum, Journal of Functional Analysis Volume 281, Issue 4, 15 August 2021, 109072
  * [19] Y.Meyer, Quasicrystals, Almost Periodic Patterns, Mean–periodic Functions, and Irregular Sampling. African Diaspora Journal of Mathematics, 13 no.1, (2012) 1-45.
  * [20] Y. Meyer, Measures with locally finite support and spectrum, Proc. Natl. Acad. Sci. USA 113(12) (2016), 31523158, DOI 10.1073/pnas.1600685113
  * [21] Y.Meyer, Guinand’s Measure are Almost Periodic Distributions. Bulletin of the Hellenic Mathematical Society, 61, (2017) 11-20.
  * [22] Y. Meyer, Global and local estimates on trigonometric sums, Trans. R. Norw. Soc. Sci. Lett. 2018(2) 1-25.
  * [23] V.P.Palamodov, A Geometric Characterization of a Class of Poisson Type Distributions, Journal of Fourier Analysis and Applications, 23, no.5, (2017) 1227–1237.
  * [24] Quasicrystals and Discrete Geometry. J.Patera,ed., Fields Institute Monographs, AMS, Providence RI, 289p.
  * [25] L.I. Ronkin Almost Periodic Distributions and Divisors in Tube Domains, Zap. Nauchn. Sem. POMI 247 (1997) 210–236 (Russian).
  * [26] W.Rudin. Functional Analysis, McGraw-Hill Book Company, New York, St.Louis, Sun Francisco, (1973), 443p.
  * [27] V.S.Vladimirov. Equations of Mathematical Physics, Marcel Dekker, Inc.,New-York, 1971, 418p.