a lacunary series
/ luh-KYOO-nuh-ree /
Most power series have terms for every power: a_0 + a_1 z + a_2 z^2 + a_3 z^3 + ... A lacunary series is one riddled with gaps — almost all the coefficients are zero, and the surviving powers are spaced ever further apart as you go out. The word lacuna means a gap or empty space, and that is exactly the feature: between one nonzero term and the next, the exponent makes a big leap. A typical shape is sum c_k z^(n_k) where the exponents n_k race off to infinity with widening gaps, for example n_k = 2^k giving z + z^2 + z^4 + z^8 + ...
The precise condition that makes the gaps 'big enough' is Hadamard's gap condition: there is a fixed ratio q greater than 1 so that each exponent is at least q times the previous one, n_(k+1) >= q n_k. Under this condition (and assuming the disk of convergence is |z| < 1, say), the Ostrowski-Hadamard gap theorem guarantees something dramatic: the boundary circle of convergence is a natural boundary. The intuition is that the gaps make the partial sums oscillate so violently near the boundary, in so many directions at once, that singularities are forced onto a dense set of boundary points, and no continuation across the circle is possible. The gaps, paradoxically, lock the function in.
Why study these? First, they are the cleanest constructions of functions with a natural boundary — when you want an explicit example of a holomorphic function that absolutely cannot be continued, a Hadamard gap series is the go-to. Second, they reveal a deep theme: the analytic continuation of a function is encoded subtly in the size AND the spacing of its Taylor coefficients, not just their size. Two series with coefficients of the same magnitude can behave totally differently at the boundary depending on whether the nonzero terms are spread out or clumped. Lacunary series also recur in harmonic analysis (lacunary Fourier series) and in the study of random-like analytic functions.
The series sum_{k>=1} z^(3^k) = z^3 + z^9 + z^27 + ... satisfies Hadamard's gap condition with ratio q = 3 (each exponent is 3 times the last). By the gap theorem its radius of convergence is 1 and the unit circle is a natural boundary: even though the series looks tame and converges nicely inside, it cannot be extended one step beyond the disk.
Exponents tripling each step satisfy the gap condition, so the unit circle becomes an impassable natural boundary.
Big gaps do not weaken the series — they imprison it. It is a common surprise that a series this sparse and well-behaved inside the disk can be utterly trapped there, unable to continue at any boundary point.