abstract We isolate a basic geometric fact for translation-type interactions on a finite interval. Let \(I_a=[-a,a]\) and let a shift by displacement \(\ell\) act on functions supported in \(I_a\). The shifted support can intersect the original window if and only if \(|\ell|\le 2a\). Thus \(2a\) is the exact maximal displacement visible inside the window. When a displacement is parameterized logarithmically by \(\ell=\log n\), the visibility condition becomes \(n\le \exp(2a)\). The result is purely geometric: it uses only the diameter of the interval and contains no auxiliary analytic or numerical assumptions. abstract

Motivation and provenance

The setting considered here is elementary, but it arises naturally in the study of finite-window constructions involving translations. A parameter \(a\) specifies a bounded interval, while a displacement is represented by a real number \(\ell\). The basic geometric question is whether a translated copy of the window can still meet the original one [krein1977,suzuki2023,coburn1969].

This note isolates that question from the additional structures in which it may occur. The key point is that the answer is determined entirely by the diameter of the window. Once the displacement is written in logarithmic coordinates, the same statement acquires the form \(n\le e^{2a}\).

The purpose of the note is to state and prove this geometric fact in its simplest form. No analytic, spectral, asymptotic, or numerical ingredient is needed.

The finite window

Fix \(a>0\) and define the closed interval \[I_a:=[-a,a].\] Its diameter is \[\operatorname{diam}(I_a)=2a.\] Let \(S_\ell\) denote translation by a real displacement \(\ell\). For definiteness, we use \[(S_\ell f)(x):=f(x-\ell).\] The only question considered in this paper is geometric: when can a translated copy of an object supported in \(I_a\) still meet the original interval?

Shifted-window overlap

The translated interval is \[I_a+\ell=[\ell-a,\ell+a].\] The relevant interaction region is therefore the intersection \[I_a\cap(I_a+\ell).\] We now state the central result.

theorem[Finite-Window Shift Theorem] For \(a>0\) and \(\ell\in\mathbb R\), the following are equivalent:

  1. \(I_a\cap(I_a+\ell)\neq\varnothing\);
  2. there exist \(x,y\in I_a\) with \(y-x=\ell\);
  3. \(|\ell|\le 2a\).

Equivalently, a displacement is visible inside the window \(I_a\) exactly when its absolute magnitude does not exceed the diameter of the window. theorem

proof Assume first that \(I_a\cap(I_a+\ell)\) is nonempty. Then there exists \(x\in I_a\) such that \(x\in I_a+\ell\). Hence \(x-\ell\in I_a\). Therefore both \(x\) and \(x-\ell\) belong to \([-a,a]\), so \[|\ell|=|x-(x-\ell)|\le \operatorname{diam}(I_a)=2a.\] Thus (i) implies (iii).

Conversely, suppose \(|\ell|\le 2a\). If \(\ell\ge0\), choose \(x=a-\ell\) and \(y=a\). Then \(x,y\in[-a,a]\) and \(y-x=\ell\). Equivalently, the intervals \([-a,a]\) and \([\ell-a,\ell+a]\) meet. The case \(\ell<0\) is identical after reflection. Hence (iii) implies (i).

The equivalence with (ii) is simply the statement that two points of \(I_a\) can differ by \(\ell\) exactly when the magnitude of that difference does not exceed the interval diameter. This completes the proof. proof

Logarithmic displacement

The theorem becomes particularly useful when the displacement is parameterized by a positive quantity \(n\) through \[\ell=\log n,\qquad n\ge1.\] Because \(\log n\ge0\) for \(n\ge1\), Theorem 1 gives the equivalent condition \[\log n\le2a.\] Since the exponential function is strictly increasing, this is equivalent to \[n\le e^{2a}.\] Thus we obtain the exact logarithmic visibility law.

corollary[Logarithmic Visibility Law] For \(n\ge1\), \[\log n\le2a\quad\Longleftrightarrow\quad n\le e^{2a}.\] No approximation is involved: the boundary is determined exactly by the diameter \(2a\) of the window. corollary

Thresholds

For each \(n\ge1\), define its opening threshold by the equality \(\log n=2a\). This gives \[a_n:=\frac12\log n.\] Then:

  • if \(a<a_n\), the shift \(\log n\) lies beyond the geometric reach of the window;
  • if \(a=a_n\), the shifted and unshifted intervals meet at a boundary point;
  • if \(a>a_n\), the two intervals overlap on a nonempty interval.

Thus each logarithmic displacement becomes geometrically accessible at its threshold \(a_n\).

A direct geometric formulation

The same statement can be expressed without introducing any operator notation. Two points \(x,y\in[-a,a]\) satisfy \[y=x+\ell\] for some \(x,y\) in the window if and only if \[|y-x|\le2a.\] Therefore every displacement realized by a pair of points in the window belongs to the interval \[[-2a,2a].\] Consequently, the exact set of visible logarithmic displacements is \[\{\log n:\ n\ge1\ \text{and}\ \log n\le2a\}.\] This is the entire content of the theorem.

Examples

The threshold law gives an immediate scale table: \[\begin{array}{c|c|c} a & 2a & e^{2a}\\ \hline 1 & 2 & 7.38906\\ 1.5 & 3 & 20.0855\\ 2 & 4 & 54.5981\\ 2.5 & 5 & 148.413\\ 3 & 6 & 403.429 \end{array}\] These numbers are illustrations only; the theorem itself is exact and does not depend on numerical evaluation.

Scope of the result

The theorem is deliberately restricted to finite-window geometry. No additional structure is required. In particular, the statement does not depend on any basis, discretization, matrix representation, asymptotic assumption, or numerical computation.

The role of the logarithm is only to provide a reparameterization of the displacement. The geometric statement itself is the diameter bound \(|\ell|\le2a\).

Conclusion

A finite interval of radius \(a\) has a finite geometric reach equal to its diameter \(2a\). Any displacement whose magnitude exceeds \(2a\) cannot connect two points of the window. When displacements are labeled logarithmically by \(\ell=\log n\), the same geometric fact becomes the exact threshold \[n\le e^{2a}.\] This theorem is the complete result of the present note.

Scope and provenance

The formulation above is self-contained. The motivation for considering finite windows together with translation parameters is rooted in the classical literature on screw functions and related continuation problems; the finite-interval framework used in this note is part of that mathematical context [krein1977,suzuki2023,coburn1969]. Theorem 1 itself is an elementary geometric statement and is proved here independently.

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suzuki2023 M. Suzuki, «Aspects of the screw function corresponding to the Riemann zeta-function,» J. London Math. Soc. 108 (2023), 1448–1487. DOI: .

coburn1969 L. A. Coburn and R. G. Douglas, «Translation operators on the half-line,» Proc. Natl. Acad. Sci. USA 62 (1969), 1010–1013. DOI: . thebibliography