MORE ON POINTS AND ARCS

VSEVOLOD F. LEV

Abstract. Let z_{1}, …, z_{N} be complex numbers, situated on the unit circle |z| = 1 so that any open arc of length φ ∈ (0, π] of the circle contains at most n of them. Write S := z_{1} + ⋯ + z_{N}. Complementing our earlier result, we show that

|S| ≤ n \frac{sin(φN/2n)}{sin(φ/2)}.

Consequently, given that |S| ≥ αN with α ∈ (0, 1], there exists an open arc of length φ containing at least

\frac{φ/2}{g^{−1}(αg(φ/2))} N

of the numbers z_{1}, …, z_{N}; here g(x) = sin x/x and g^{−1} is the function, inverse to g on the interval 0 < x ≤ π.

Let 𝕌 := {z ∈ ℂ: |z| = 1} and suppose that for integers N ≥ n ≥ 1 and real φ ∈ (0, π], the numbers z_{1}, …, z_{N} ∈ 𝕌 have the property that any open arc of 𝕌 of length φ contains at most n of them. Extending a well-known lemma of Freiman [F62, Lemma 1], we showed in [L05] that, writing S := z_{1} + ⋯ + z_{N}, one has

|S| ≤ 2n − N + 2(N − n) cos(φ/2);  (1)

thus, if |S| ≥ αN with α ∈ [0, 1], then there is an open arc of 𝕌 of length φ containing at least

\frac{α + 1 − 2 cos(φ/2)}{2(1 − cos(φ/2))} N  (2)

of the numbers z_{1}, …, z_{N}. Estimate (1) is sharp in the range N/2 ≤ n ≤ N: equality is attained, for instance, if 2n−N of the numbers z_{1}, …, z_{N} equal 1, and the remaining 2N − 2n of them are evenly split between exp(iφ/2) and its conjugate exp(−iφ/2). Accordingly, the bound (2) is sharp if α ≥ cos(φ/2). Indeed, if in this case n is the smallest integer, greater than or equal to the expression in (2), then N/2 ≤ n ≤ N and the configuration just described provides an example of z_{1}, …, z_{N} with |S| ≥ αN (as it follows from a brief computation) and no open arc of length φ containing more than n of the numbers z_{1}, …, z_{N}.

On the other hand, (1) and (2) can be far from sharp if n < N/2 and α < cos(φ/2), respectively. For instance, straightforward averaging shows that there is an arc of length φ, containing at least (φ/2π)N of z_{1}, …, z_{N}. This nearly trivial bound is

2 VSEVOLOD F. LEV

better, than (2), if

α < 1 − (2 − φ/π)(1 − cos(φ/2));

that is, when both φ and α are small. Below we establish an estimate which remains reasonably sharp for small values of n and α and, in particular, is better than the trivial estimate for the whole range of parameters.

Theorem 1. Let N and n be positive integers and let φ ∈ (0, π]. Suppose that the numbers z_{1}, …, z_{N} ∈ 𝕌 have the property that any open arc of 𝕌 of length φ contains at most n of them. Then, writing S := z_{1} + ⋯ + z_{N}, we have

|S| ≤ n \frac{sin(φN/2n)}{sin(φ/2)}.

For the rest of the note, we write g(x) := sin x/x and denote by g^{−1} the function, inverse to g on the interval 0 < x ≤ π. Notice, that g^{−1} is defined and monotonically decreasing on [0, 1).

Corollary 1. Let N be a positive integer and let φ ∈ (0, π]. Suppose that z_{1}, …, z_{N} ∈ 𝕌 and write S := z_{1} + ⋯ + z_{N}. If |S| ≥ αN with α ∈ (0, 1], then there is an open arc of 𝕌 of length φ containing at least

\frac{φ/2}{g^{−1}(αg(φ/2))} N

of the numbers z_{1}, …, z_{N}.

To deduce Corollary 1 from Theorem 1 observe that if N, φ, z_{1}, …, z_{N}, S, and α are as in the corollary, and if n is the largest number of points among z_{1}, …, z_{N} on an open arc of length φ, then

αN ≤ n \frac{sin(φN/2n)}{sin(φ/2)}

by Theorem 1, whence

α \frac{sin(φ/2)}{φ/2} ≤ \frac{sin(φN/2n)}{φN/2n}.

Equivalently, αg(φ/2) ≤ g(φN/2n), and the assertion follows by applying g^{−1} to both sides.

Note that the bound of Corollary 1 is attained if α = sin(dφ/2)/(d sin(φ/2)), where 1 ≤ d ≤ 2π/φ is an integer and N is divisible by d. For, set in this case n := N/d and consider a d-term geometric progression with the ratio exp(iφ), situated on 𝕌. Placing exactly n points at each term of this progression, we obtain a system of

MORE ON POINTS AND ARCS 3

N complex numbers such that no open arc of 𝕌 of length φ contains more than n = (φ/2)N/g^{−1}(αg(φ/2)) of them, while their sum equals αN in absolute value.

Finally, we notice that the bound of Corollary 1 is better than (2) for all α and φ such that α ≤ cos(φ/2); we omit the (rather tedious) verification.

The remainder of the note is devoted to the proof of Theorem 1. We start with a lemma.

Lemma 1. Suppose that the function f ∈ L^{1}[−π, π] attains values in the interval [0, 1]. If ∫_{−π}^{π} f(θ) dθ = 2c (with a real c), then

∫_{−π}^{π} f(θ) cos θ dθ ≤ 2 sin c.

Proof. Let I_{c} denote the indicator function of the interval [−c, c]. Then

f(θ)(cos θ − cos c) ≤ I_{c}(θ)(cos θ − cos c)

for all θ ∈ [−π, π], and it follows that

∫_{−π}^{π} f(θ) cos θ dθ = ∫_{−π}^{π} f(θ)(cos θ − cos c) dθ + 2c cos c
≤ ∫_{−c}^{c}(cos θ − cos c) dθ + 2c cos c
= 2 sin c.
□

Remark. The estimate of Lemma 1 is attained for f = I_{c}. Thus, what the lemma actually says is that for a given value of ∫_{−π}^{π} f(θ) dθ, the integral ∫_{−π}^{π} f(θ) cos θ dθ is maximized if f is concentrated around 0 (where cos θ is maximal).

Proof of Theorem 1. Without loss of generality we can assume that S is real.

For θ ∈ [−π, π], let K(θ) denote the number of those indices j ∈ [1, N] such that there is a value of arg z_{j} which is within less than φ/2 from θ; with a little abuse of notation, we can write

K(θ) := #{j ∈ [1, N]: | arg z_{j} − θ| < φ/2}.

Notice that K(θ) is piecewise continuous and attains values in [0, n]. Furthermore, it is readily verified that

∫_{−π}^{π} K(θ) dθ = φN,

and applying Lemma 1 to the function f(θ) := K(θ)/n we conclude that

∫_{−π}^{π} K(θ) cos θ dθ ≤ 2n sin \frac{φN}{2n} .

To complete the prove we observe that the integral in the left-hand side is

ℜ( ∑_{j=1}^{N} ∫_{arg z_{j}−φ/2}^{arg z_{j}+φ/2} exp(iθ) dθ) = ℜ( ∑_{j=1}^{N} z_{j} · 2 sin(φ/2)) = 2S sin(φ/2).

□

References

[F62] G.A. Freiman, Inverse problems of additive number theory, VII. On addition of finite sets, IV, Izv. Vyss. Ucebn. Zaved. Matematika 6 (31) (1962), 131–144.

[L05] V.F. Lev, Distribution of points on arcs, INTEGERS: The Electronic Journal of Combinatorial Number Theory 5(2) (2005), A11, 8 pp. (elctronic).

Department of Mathematics, The University of Haifa at Oranim, Tivon 36006, Israel

  E-mail address: seva@math.haifa.ac.il