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In mathematics, a bounded sequence {s1, s2, s3, …} of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that interval. Such sequences are studied in Diophantine approximation theory and have applications to Monte Carlo integration.

Definition

A bounded sequence {s1, s2, s3, …} of real numbers is said to be equidistributed on an interval [ab] if for any subinterval [cd] of [ab] we have

(Here, the notation |{s1,…,sn }∩[c,d]| denotes the number of elements, out of the first n elements of the sequence, that are between c and d.)

For example, if a sequence is equidistributed in [0, 2], since the interval [0.5, 0.9] occupies 1/5 of the length of the interval [0, 2], as n becomes large, the proportion of the first n members of the sequence which fall between 0.5 and 0.9 must approach 1/5. Loosely speaking, one could say that each member of the sequence is equally likely to fall anywhere in its range. However, this is not to say that {sn} is a sequence of random variables; rather, it is a determinate sequence of real numbers.

Discrepancy

We define the discrepancy D(N) for a sequence {s1, s2, s3, …} with respect to the interval [ab] as

A sequence is thus equidistributed if the discrepancy D(N) tends to zero as N tends to infinity.

Equidistribution is a rather weak criterion to express the fact that a sequence fills the segment leaving no gaps. For example, the drawings of a random variable uniform over a segment will be equidistributed in the segment, but there will be large gaps compared to a sequence which first enumerates multiples of ε in the segment, for some small ε, in an appropriately chosen way, and then continues to do this for smaller and smaller values of ε. See low-discrepancy sequence for stronger criteria and constructions of low-discrepancy sequences for constructions of sequences which are more evenly distributed.

Equidistribution modulo 1

The sequence {a1, a2, a3, …} is said to be equidistributed modulo 1 or uniformly distributed modulo 1 if the sequence of the fractional parts of the an's, (denoted by an−⌊an⌋)

is equidistributed in the interval [0, 1].

Examples

  • The sequence of all multiples of an irrational α,
0, α, 2α, 3α, 4α, …

is uniformly distributed modulo 1:[1] this is the equidistribution theorem.

  • More generally, if p is a polynomial with at least one irrational coefficient (other than the constant term) then the sequence p(n) is uniformly distributed modulo 1: this was proven by Weyl and is an application of van der Corput's difference theorem.[2]
  • The sequence log(n) is not uniformly distributed modulo 1.[1]
  • The sequence of all multiples of an irrational α by successive prime numbers,
2α, 3α, 5α, 7α, 11α, …

is equidistributed modulo 1. This is a famous theorem of analytic number theory, proven by I. M. Vinogradov in 1935.

van der Corput's difference theorem

A theorem of Johannes van der Corput[4] states that if for each h the sequence sn+h−sn is uniformly distributed modulo 1, then so is sn.[5][6]

A van der Corput set is a set H of integers such that if for each h in H the sequence sn+h−sn is uniformly distributed modulo 1, then so is sn.[6]

Properties

The following three conditions are equivalent:

  1. {an} is equidistributed modulo 1.
  2. For every Riemann integrable function f on [0, 1],
  1. For every nonzero integer k,

The third condition is known as Weyl's criterion. Together with the formula for the sum of a finite geometric series, the equivalence of the first and third conditions furnishes an immediate proof of the equidistribution theorem.

Metric theorems

Metric theorems describe the behaviour of a parametrised sequence for almost all values of some parameter α: that is, for values of α not lying in some exceptional set of Lebesgue measure zero.

  • For any sequence of distinct integers bn, the sequence {bn α} is equidistributed mod 1 for almost all values of α.[7]
  • The sequence {αn} is equidistributed mod 1 for almost all values of α > 1.[8]

It is not known whether the sequences {en} or {πn} are equidistributed mod 1. However it is known that the sequence {αn} is not equidistributed mod 1 if α is a PV number.

Well-distributed sequence

A bounded sequence {s1, s2, s3, …} of real numbers is said to be well-distributed on [ab] if for any subinterval [cd] of [ab] we have

uniformly in k. Clearly every well-distributed sequence is uniformly distributed, but the converse does not hold. The definition of well-distributed modulo 1 is analogous.

Sequences equidistributed with respect to an arbitrary measure

For an arbitrary probability measure space , a sequence of points is said to be equidistributed with respect to if the mean of point measures converges weakly to :

It is true, for example. that for any probabilistic borel measure on a separable, metrizable space, there exists an equidistributed sequence (with respect to the measure).

See also

References

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  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

Further reading

  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links



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  1. 1.0 1.1 Kuipers & Niederreiter (2006) p. 8
  2. Kuipers & Niederreiter (2006) p. 27
  3. Kuipers & Niederreiter (2006) p. 127
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  5. Kuipers & Niederreiter (2006) p. 26
  6. 6.0 6.1 Montgomery (1994) p.18
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    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

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    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010