The Tu-Deng conjecture is a bound on pairs of binary expansions with prescribed modular sum. For integers 
and 
, let 
be the number of 1s in the binary expansion of 
, and define
 |
The conjecture asserts that 
(Cusick 2026).
Cusick (2026) gives the equality characterization following proofs of the bound. If the 
-bit
expansion of 
has 
zeros, then 
iff each cyclic distance between consecutive zeros is at least
,
where the distance includes the final zero. In particular, equality requires 
. The number of equality words
on labeled cyclic positions with exactly 
zeros is
 |
when 
,
and is zero otherwise (Cusick 2026).
